Possibly, if the current layout refers to table 2 in Layout setup... and displays related data from table 1. It may look to the user like a delete records action will delete records from table 1 instead of table 2. Since this action will really delete from table 2, the privilege settings will block it.
Another option: Do you have a relationship between the tables set so that records in table 2 will be deleted when a related record in table 1 is deleted? I haven't tested this, but that might explain the issue.
This led me to the right direction. I forgot about Table 3 -- a table I had started to work with and then discontinued... but I left the relationship intact. Once I removed that, all is well.