TL;DR: probably not, but we can't really tell for sure
Prologue: This is a hard question, and unanswerable at this point
This is a problematic question to answer. It's a very hard question to answer, even if the relics were given for scientific examination, which doesn't happen. More over, the answer to this question relies on the answers to several other question, like "Was there a real person that would match the description of Jesus", "Was this man crucified" and "Was he crucifies on a cross, and not on another Crucifixion devise, like a T shaped stand or a pole?" Questions to which we do not have a proven scientific answer as of this date.
Obviously if Jesus didn't exist, or if he wasn't crucified on a cross a "True Cross" can not exist. And those are question which we can't answer with a positive "Yes".
First part: what question can we answer
Obviously that the best way to examine the claims is to take each piece of alleged wood and check from what time it is, and from what wood. Pieces that would be to young or made of wood that wasn't used by the Romans for crucifiction would immediately be rejected and proven as forgery. But the relics aren't put under this test, and even if some will pass this, this doesn't mean that they are of the true cross, it would just mean that they were cut before 30 AD and are of the same wood that the Romans used for crucifictions1.
We can look at the "evidence" that are presented to us, mainly the story of how the true cross was found, and judge if the evidence present at that occasion were enough to assume that the cross that was found is indeed the true cross.
Second Part: How the "True Cross" was found
The true cross doesn't have a well documented chain of custody. Here is what the catholic church says about its origin (Source: The Catholic Encyclopedia):
The second event was of even greater importance. In the year 326 the mother of Constantine, Helena, then about 80 years old, having journeyed to Jerusalem, undertook to rid the Holy Sepulchre of the mound of earth heaped upon and around it, and to destroy the pagan buildings that profaned its site, Some revelations which she had received gave her confidence that she would discover the Saviour's Tomb and His Cross. The work was carried on diligently, with the co-operation of St. Macarius, bishop of the city. The Jews had hidden the Cross in a ditch or well, and covered it over with stones, so that the faithful might not come and venerate it. Only a chosen few among the Jews knew the exact spot where it had been hidden, and one of them, named Judas, touched by Divine inspiration, pointed it out to the excavators, for which act he was highly praised by St. Helena. Judas afterwards became a Christian saint, and is honoured under the name of Cyriacus. During the excavation three crosses were found, but because the titulus was detached from the Cross of Christ, there was no means of identifying it. Following an inspiration from on high, Macarius caused the three crosses to be carried, one after the other, to the bedside of a worthy woman who was at the point of death. The touch of the other two was of no avail; but on touching that upon which Christ had died the woman got suddenly well again. From a letter of St. Paulinus to Severus inserted in the Breviary of Paris it would appear that St. Helena. herself had sought by means of a miracle to discover which was the True Cross and that she caused a man already dead and buried to be carried to the spot, whereupon, by contact with the third cross, he came to life. From yet another tradition, related by St. Ambrose, it would seem that the titulus, or inscription, had remained fastened to the Cross.
See that bolded part (bolded by me)? Even the Catholic church isn't sure of how the true cross was identified. If we look in the Wikipedia Page we see that there are several different descriotion of the erection of the Church, stories that differ by important details including whether the cross was found, the artifacts found with it, and who found it:
According to Eusebius
Eusebius of Caesarea, in his Life of Constantine, describes how the site of the Holy Sepulchre, originally a site of veneration for the Christian community in Jerusalem, had been covered with earth and a temple of Venus had been built on top — although Eusebius does not say as much, this would probably have been done as part of Hadrian's reconstruction of Jerusalem as Aelia Capitolina in 135, following the destruction during the Jewish Revolt of 70 and Bar Kokhba's revolt of 132–135. Following his conversion to Christianity, Emperor Constantine ordered in about 325–326 that the site be uncovered and instructed Saint Macarius, Bishop of Jerusalem, to build a church on the site. In his Life of Constantine, Eusebius does not mention the finding of the True Cross.
According to Socrates Scholasticus
Socrates Scholasticus (born c. 380), in his Ecclesiastical History, gives a full description of the discovery that was repeated later by Sozomen and by Theodoret. In it he describes how Saint Helena, Constantine's aged mother, had the temple destroyed and the Sepulchre uncovered, whereupon three crosses and the titulus from Jesus's crucifixion were uncovered as well. In Socrates's version of the story, Macarius had the three crosses placed in turn on a deathly ill woman. This woman recovered at the touch of the third cross, which was taken as a sign that this was the cross of Christ, the new Christian symbol. Socrates also reports that, having also found the nails with which Christ had been fastened to the cross, Helena sent these to Constantinople, where they were incorporated into the emperor's helmet and the bridle of his horse.
According to Sozomen
Sozomen (died c. 450), in his Ecclesiastical History, gives essentially the same version as Socrates. He also adds that it was said (by whom he does not say) that the location of the Sepulchre was "disclosed by a Hebrew who dwelt in the East, and who derived his information from some documents which had come to him by paternal inheritance" (although Sozomen himself disputes this account) and that a dead person was also revived by the touch of the Cross. Later popular versions of this story state that the Jew who assisted Helena was named Jude or Judas, but later converted to Christianity and took the name Kyriakos.
According to Theodoret
The proving of the True Cross, Jean Colombe in the Très Riches Heures.
Theodoret (died c. 457) in his Ecclesiastical History Chapter xvii gives what had become the standard version of the finding of the True Cross:
When the empress beheld the place where the Saviour suffered, she immediately ordered the idolatrous temple, which had been there erected, to be destroyed, and the very earth on which it stood to be removed. When the tomb, which had been so long concealed, was discovered, three crosses were seen buried near the Lord's sepulchre. All held it as certain that one of these crosses was that of our Lord Jesus Christ, and that the other two were those of the thieves who were crucified with Him. Yet they could not discern to which of the three the Body of the Lord had been brought nigh, and which had received the outpouring of His precious Blood. But the wise and holy Macarius, the president of the city, resolved this question in the following manner. He caused a lady of rank, who had been long suffering from disease, to be touched by each of the crosses, with earnest prayer, and thus discerned the virtue residing in that of the Saviour. For the instant this cross was brought near the lady, it expelled the sore disease, and made her whole.
With the Cross were also found the Holy Nails, which Helena took with her back to Constantinople. According to Theodoret, "She had part of the cross of our Saviour conveyed to the palace. The rest was enclosed in a covering of silver, and committed to the care of the bishop of the city, whom she exhorted to preserve it carefully, in order that it might be transmitted uninjured to posterity."
Another popular ancient version from the Syriac tradition replaced Helena with a fictitious first-century empress named Protonike.
Historians consider these versions to be apocryphal in varying degrees. It is certain, however, that the Basilica of the Holy Sepulchre was completed by 335 and that alleged relics of the Cross were being venerated there by the 340s, as they are mentioned in the Catecheses of Cyril of Jerusalem
Here are some problems with the story:
One version of the story claims that the Jews hid the true cross to prevent christian from worshiping it. If you want to destroy the relics of another religion, you just destroy it, if the Jews did get hold of the true cross and wanted to prevent the christian from worshiping it, they would have simply burn it, not hide it. Just like the many times that this happen through history, by the Taliban, by Christians, Romans and many others.
The cross was supposedly dug out of the ground approximately 300 years after its use, and the only proof for its authenticity was a miracle, that wasn't recorded in any other contemporary source.
Even though the cross could allegedly heal the dying it was never recorded to be used for this purpose except for the one time when it was identified.
A contemporary source describing the building of the church and destroying of the temple that stood in its place, doesn't bother to include that a very important relic was found, and it completely healed a dying person.
Conclusion: the available "proofs" for the relics to be real are not enough
After examining the origin story, a skeptic must admit that the proofs presented for the validity of the relic are very slim, and do not hold under scientific scrutiny. However this isn't conclusive.
In order to truly falsify each relic, each of them needs to be examined, firstly to be of the right parameters of age type and origin, and secondly for its miraculous healing powers.
1: Unless that piece of wood could raise the dead or heal the sick (like cure AIDS and terminal stage cancer, or regrow limbs, not relieve mild headache), then it's (in my eyes at least) a sufficient evidence that Jesus's story is true, he was/is the son of God, and this is a piece of the true cross.