Edsheeranalbumparty (great name btw), there is an emerging field that deals with the nonspecific effects of vaccines. I've linked to a couple of reviews below explaining some of what is known so far in an accessible way. Most vaccine schedules are based on the assumption that a particular vaccine (say Measles) will protect against that particular disease and do nothing else. There are now many studies showing that this is not the case, and that each vaccine has both a specific effect (so against Measles in case this case) but also a nonspecific effect on the immune system in general. This can be beneficial or detrimental, depending on the vaccine, but generally speaking live vaccines are 'good' and inactive ones are 'bad' for the immune system as a whole. For instance, there is a fair bit of evidence from developing countries that DTP increases all cause mortality (while lowering the rates of the diseases targeted), and a similar result has now been shown in Denmark, although in this case the outcome was an increase in respiratory infections.
There is also evidence that age, sex, and nutritional status can alter the effect of vaccines, which is also not acknowledged in the current schedule. A logical extension of that is that individual genetics may have an effect, which we're a long way from understanding.
I have also seen studies suggesting that the order in which vaccines are given, and whether more than one are given at once, can impact outcomes, but I'll have to dig those out.
The problem is that this entire field is not yet on the radar of the people making the decisions, and the assumption that you can give essentially unlimited vaccines simultaneously with no off target effects tends to prevail. Similarly, vaccine trials tend to only look at protection against targeted diseases as outcomes, and rarely look at infection rates as a whole or the development of allergies. As far as I'm aware there are no longer term studies into the effect of the current schedule (up to four at once, with Calpol in some cases) on the immune system as a whole. Our system is very reactive and only changes guidelines when there is clear evidence of a problem, so it is likely that this might remain the case for some years. This doesn't, however, mean it's the best medically. There is an awful lot that is still poorly understood about how the immune system works, and my argument is that we should exercise caution when 'modifying' it to cause as little disruption as possible. I think there is a clear need to protect against Measles, Polio etc, and so support vaccine use in principle, but I think in practice a safer approach would be to spread them out and lessen the 'assault' on the immune system at any one time.
www.ncbi.nlm.nih.gov/pmc/articles/PMC3689344/
www.cell.com/trends/immunology/abstract/S1471-4906(13)00058-6?_returnURL=http%3A%2F%2Flinkinghub.elsevier.com%2Fretrieve%2Fpii%2FS1471490613000586%3Fshowall%3Dtrue