I think the tests are reliable but overly optimistic. I have not done any testing with Ancestry but I have done it with FamilyTree DNA. My results are similar to the poster's wife: about 350 3rd-5th cousins and about 3,500 total matches.
I think there are two root causes.
1. Ashkenazi jews are very inbred. There were a lot of marriages between cousins, meaning that we share more common DNA than more genetically diverse populations. The DNA testing firms should probably set higher matching thresholds for these inbred (endogamous) populations.
2. I have a feeling that a disproportionate number of those testing DNA are Jewish.