Index a field containing people's names
Try building a GIN index with gin_trgm_ops instead of GiST.  I've never found GiST useful for trgm indexes.

That said, if your are failing with that message, you probably have some weird data in your table that should be cleaned up. Can you figure which row is causing the problem?  (If not, start with the longest records first as the most likely suspects)