Fast Compression backup errors with warning message:
Failed: Does not match LSN of full backup
FastCompression backup sequence has failed LSN verification
Executing SqlLiteSpeed.exe: Write new full backup "New file name here"
The LSN of the database does not match the LSN of the last full backup.
This may be due to a restore of the database to an earlier point in time, an external Full backup (not done by the Fast Compression plan)
Fast Compression resolves the issue by creating a Full backup file, using the current checkpoint.
Also see http://technet.microsoft.com/en-us/library/ms189573.aspx for more details on database checkpoints.