If you have your website on a VPS, log in with putty (or better - winscp - so you can visualize all files) cd to /tmp folder and remove all files.
Or if your website is on a shared hosting, ask your provider to empty their /tmp directory.
This error appears on 3 different cases:
1. The server's hard drive is full
2. The /tmp directory is full
3. There are not sufficient privileges to write to /tmp directory.
Absolutely CORRECT! Also just an advisory warning. If you are on a Windows server, the more likely cause of this error is Permissions. I have had this happen to me so many times. And depending on what administration platform your server is running you may or may not be able to change the permissions in IIS or in general using RDC/P or Putty. PLESK for example will overwrite any permissions changes you make via command line or IIS. If you don't go and change the permissions specifically in the Platform it wont stick and you will find that you can upload files for a while. But your permissions will eventually drop back off.