As a parent, I often hear people say that they think it's bad for their children to have several vaccines in one go. Some parents pay for some vaccines to be given separately, and I wondered if there is any evidence that it makes a difference.
This argument, "Too many, too soon", is very common among groups opposing vaccines and promoting alternative cures. The basic argument is that the number of vaccines given to children increased substantially and that this "overloads" the immune system and causes illnesses, e.g. autism. I've heard this claim almost exclusively in connection with autism, so I'll focus on that in my answer.
The number of vaccines you can find in the recommended vaccine schedule for the United States published by the CDC.
But in the end the number of vaccines is nothing compared to the number of antigens children will encounter naturally. Paul Offit said the following about that subject:
A baby’s body is bombarded with immunologic challenges—from bacteria in food to the dust they breathe. Compared to what they typically encounter and manage during the day, vaccines are literally a drop in the ocean”, and Dr. Offits studies theoretically show an infant could handle up to 100,000 vaccines at one time … safely (quoted from SBM)
There is a study published in the Journal of Pediatrics titled "On-time Vaccine Receipt in the First Year Does Not Adversely Affect Neuropsychological Outcomes" that directly refutes the claim that too many vaccines cause neurodevelopmental disorders in children. From that study:
Timely vaccination during infancy has no adverse effect on neuropsychological outcomes 7 to 10 years later. These data may reassure parents who are concerned that children receive too many vaccines too soon.
On rereading the question I see that I misread it somwhat, but I think the evidence also holds true for the case of simultaneous vaccinations, and the CDC agrees with that.
There seems to be plenty of anecdotal evidence (currently the 4th google result for "multiple vaccines") to confirm this, but according to the CDC, there have been studies which have found no adverse health effects associated with multiple simultaneous vaccinations (emphasis mine):
The available scientific data show that simultaneous vaccination with multiple vaccines has no adverse effect on the normal childhood immune system. A number of studies have been conducted to examine the effects of giving various combinations of vaccines simultaneously. These studies have shown that the recommended vaccines are as effective in combination as they are individually, and that such combinations carry no greater risk for adverse side effects.
The CDC also mentions that separating vaccinations is more likely to cause discomfort to the child, which makes sense-- less stress to get all of them done in one visit, I would think.
The CDC does not cite any of these studies on that page, but here's a study that I found. This is the summary (again, emphasis added):
Current studies do not support the hypothesis that multiple vaccines overwhelm, weaken, or "use up" the immune system. On the contrary, young infants have an enormous capacity to respond to multiple vaccines, as well as to the many other challenges present in the environment. By providing protection against a number of bacterial and viral pathogens, vaccines prevent the "weakening" of the immune system and consequent secondary bacterial infections occasionally caused by natural infection.
There seems to be some criticism of such studies. This criticism in particular (not all criticism) comes from the National Health Federation, which has a history of promoting claims that vaccines are dangerous, so I'm inclined to disregard it.
So, in summary, no, studies have shown that it is not dangerous to have several vaccines at the same time.