1. Check available hard drive space on the server drives.
2. Also, if you are running databases in full recovery mode, a database backup does not mark the log file for reuse. Assuming they are set to automatically grow, they will consume all available hard drive space.
At my client, the application slowed when the log file exceeded 45 GB and eventually stopped working because to add 10% (5 GB) to the log file took longer than the application's SQL connection timeout setting.
Usually, log files go to the same folder but it is possible to override the location on a log file by log file basis. Track them down and see how big the log files are.
3. Is it one database that is slow or all of them? If it's a single database, it probably needs its transaction log dealt with, dbcc shrinkdatabase, dbcc reindex, etc. Database Maintenance plans are a handy way to do those tasks. If it's the entire server, it could be free drive space, log file sizes, size of tempdb, etc.
4. Have you added more users, applications, additional server load, etc.? i.e. Do you have enough CPU and RAM for the task at hand?
5. Reboot the server.