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:
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:
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 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):
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.