As you suggest, there might be something wrong with the tape media, but I can't remember having ever encountered a "dirty tape" before.* Typically, faulty tapes will result in I/O errors that are detected at the hardware or software level. I would suggest that you contact both the drive and tape manufacturer to see if they have encountered similar issues in the past and if so, what their fix was.
Hopefully, you've followed backup best practices and you have more than one copy of your backup data for disaster recovery purposes. Using an offsite copy of your backup data should either resolve the issue or point to a hardware or software failure if the condition persists.
*In my original response, I mentioned that I could not remember having ever encountered a "dirty tape" before. That said, because I have not encountered this problem before does not mean it is not possible. A reader who is in the tape cleaning business was kind enough to inform me that this problem can surface from time to time and is not that unusual. It is apparently a result of "environmental and physical stress under which the tape is expected to perform," which can apparently be resolved through a tape cleaning process. The reader works for a company called Bow Industries that offers tape cleaning services.
This was first published in May 2008