Backup suddenly is taking too long to complete

There's basically 4 things involved with backup speed:

1. Reading from the data files - you can test this by doing a backup to NUL:, as in this example, which won't break your log-backup chain for full recovery model databases:

   BACKUP DATABASE xyz TO DISK = 'NUL:' WITH COPY_ONLY;
   

2. Compressing the data - watch CPU pressure during the backup.

3. Getting it through the network path to its final location (optional, only for network backups) - test this with a network file copy.
4. Writing it to the disk on its final target location - test this with something like CrystalDiskMark or DiskSpd, but has to be done locally (not over a network pipe, since you want to avoid #3 affecting your results).

Third party backup products like Quest LiteSpeed, Idera SQLsafe, and Red Gate SQL Backup (disclaimer: I'm independent, but I've worked with all of those companies in the past) have wizards that will test each component of that, and then tell you where the bottleneck is. If you want to find it yourself, you're going to have to test each of those components, and then find the weakest link in the chain (the lowest throughput).

For example, if other SQL Servers are writing to the same target location, they may have suddenly had an increase of data or added more servers, thereby slowing down the target.