Overwhelming the system generally gets displayed in an allergic reaction however. Maybe not true anaphylaxis but generally speaking, after a round or two of vaccines, the body should have less of a reaction to known pathogens and a quicker response time to them so that something new would have a slower response.
Animals are individuals, but if you think about them going around in the world from day to day, just as we do, their systems in theory will always be dealing with foreign invaders. Think of all the variants on the common cold, the fact that "kennel cough" is technically a complex of viruses and potential bacteria that can cause cold symptoms in dogs. Your body likely deals with different bacteria day in and day out, either picked up from wherever they may go, something they potentially ingest. If you go with that theory, your pet is constantly having to fight off something, so the chances of them handling a few pathogens at once in a vaccine should in general not cause a nasty reaction or be horribly overwhelming.