Heya eeek! Admittedly, I'm running a bit blind here, so if you want to shoot me your account info I can check for more details, but based just on what you've described, there is a good chance it IS a problem with the modem. After all, the modem is in sync the whole time, but it doesn't seem to be communicating - so it's probably not a problem with the line itself, but one of the pieces of equipment at either end. So, unless your DSLAM in the Central Office is having trouble (which is a possibility, but very rare), then it's probably the modem (a much more likely possibility).
It is also theoretically possible that it's your computer or router that is having trouble, I suppose. If you've got another computer you could try connecting it directly to the modem overnight and see if you're still having the same issue.