Try rebooting (turn off, and then turn back on) your phone, if you are using a phone app.
If you are using a PC, and a reboot does not help, scan for viruses and system issues. A corrupted OS will cause problems like you describe.
I do not know if their is a download limit, but DC Universe does advertises unlimited service.
You can also delete downloads to free up room, if that is helpful
Depends what you’re downloading too. Movies and Shows have limited caps on downloads, and you can only have so many comics offline at a time.