SMS messages are stored in a database on the device, in non-volatile memory. If you don't delete them, the memory gets full and you can't receive any more.
Go to the Play Store and install the app SMSBackup. It's a free, ad-supported app. You can use it to download and delete the messages off your phone. The downloaded messages are saved to a directory on the SD card. You can delete by date ranges, so you can leave recent ones on the phone.
Once you free up some room, that should get the messages you're waiting for. If they don't arrive, try turning wi-fi off and use the carrier's signal. This sometimes causes delivery issues. If that doesn't work, call the carrier.