In theory, I agree with the vet who said he needs to have them every three weeks for three months because they were given incorrectly before. This could be considered correct, however, there are ways around it, so it isn't necessary. If the vaccines were given once a week for three weeks straight, they would have probably negated each other. If they were given starting at five weeks, maternal antibodies would've interfered.
If this were my dog I would probably give one more set after 16 weeks and do titers three weeks later to see if they have taken. Actually, this is what titers are for.
You could just do titers now, but since they were all given so young, as sad as it is, I think I'd give another set (maybe even two if the ones already given were totally ineffective).