Groups and smart groups are not calculating their contact membership list and contact counts correctly

I'm not sure how to reproduce this, but we are testing a migration from the 4.6.x series to version 4.7.28. We've performed other migrations in the past. I've tried clicking on "Update Smart Group Counts" on the groups page, and running the "Rebuild Smart Group Cache" cron job manually. The first operation silently does nothing apparent, while the second method returns "Parameters parsed (and passed to API method): a:1:{s:7:"version";i:3;} Full message: Finished execution of Rebuild Smart Group Cache with result: Success (1)". The civicrm_group_contact_cache table is empty. All of the groups show counts as "unknown" or the same very high number. When I look at the groups' contacts, they contain the same very high number of contacts. If I "edit the search criteria" or create a new smart group, then I get a sensible number of contacts in that group. If I then save that query as a group, the same issue appears for that group when I visit that group from the "manage groups" page. When using the test group I created, the "new mailing" feature estimates no recipients for that group.