Recently on Monday, my mother had an issue with no mail coming into her Gmail inbox. She had been using Godaddy for hosting her email servers. It was weird that her google Gmail had about 100 GB left of storage, so it wasn't full. And Godaddy's Email service had the emails ready it just wasn't importing into Gmail.
After a bit of googling, I couldn't figure it out, but I noticed a lot, and I mean a LOT of emails in the main inbox in GoDaddy's email service, so I realized maybe it was full and needed clearing. So I added a 2018 folder and started moving emails from 2018 to it; (it doesn't matter what year I just liked to keep it organized). After about ten thousand moved, I went back to Gmail and clicked the "Check Mail Now" button, fetched 132 emails! To the left side of the menu, folders called "Archives" of 2015, 2016, and 2017.
Conclusion:
If you run into the same problem, create a folder and move older emails into that folder. I think it wasn't importing because there is a limit that the Godaddy servers can hold. There were a LOT of emails, so the limit must be pretty high. This is extremely specific to happen like this, so it makes sense I couldn't find any forums on it.