I was having the same problem earlier. My mailbox said it was 7/10 full, but I was not receiving mail my husband sent to my toon. I deleted one of the old items, and then a new one popped in.
I had a bunch of npc mail still in my inbox, so I deleted all of those, then the ones my husband had sent me as a test all came through.
Apparently npc mail is somehow affecting the count, even though it doesn’t reflect it in the count at the top of the mailbox.