If I have a patent and someone else had one before mine can they claim patent infringement if the USPTO examiners cited theirs?

Lots of ambiguities and left-out facts in your question.

I'm going to assume that the other person owns Design Patent A and you own Design Patent B. I also assume (because you said it was so) that the Patent Office considered Design Patent A when examining the patent application you filed which matured in your Design Patent B.

So, yes, according to the Patent Office your design is patentably distinct from Design Patent A. Your patent is presumptively valid.

A court may still, however, invalidate the patent if it's shown -- by clear and convincing evidence -- that another design renders yours unpatentable (even if that design was considered by the Patent Office when examining your patent application). Also, be aware that sometimes a product that's supposed to have the shape that's disclosed in a Design Patent does not always have that shape. Which means it could infringe on a different patent that discloses a similar-appearing design.

You really need to speak with an intellectual property litigator. Take the allegation of infringement seriously. Good luck.

In addition to the information already provided, I think the simply answer to your question is that ... Even though the USPTO granted you a patent and considered the earlier patent, your design might still infringe on the other design. The USPTO only considers novelty, an non-obviousness, NOT infringement. Consider the following example...

If the previous patent claims features A, B, and C and you patent claims A, B, C, and D, you would be infringing the earlier patent because your design incorporates the claimed elements A, B, C. However, because you disclosed feature D, you patent has novelty over the prior patent and therefore would be eligible to receive a patent from the USPTO.

To add to the good explanation of why you can get a patent but could still infringe someone else's patent, you stated that the unique parts of your product are very different when compared to the parts in the other product. If your unique parts are truly not the same or equivalent to parts claimed by the other party, and the rest of what the other party claims are not found in your product, there should be no infringement.

Since the other party is accusing you of infringement, there has to be a comparison of your product with the other party's patent claims. In general, if the people accusing you of patent infringement can prove by clear and convincing evidence that your product has all of the features of at least one of their patent claims, a court can find that there is patent infringement. An important thing to remember is that an actual product has to be compared against patent claims; just because two issued patents are similar doesn't mean patent infringement exists. Sometimes patent holders go on "fishing expeditions" when they see patents issued that are similar to their patent. Such paper comparisons wouldn't hold up in court, but they may work to scare people into licenses they would not otherwise not consider. Bottom line: anyone asserting infringement should be made to show how a potentially infringing product contains all of the claim elements of one of their patent claims.