Could this be a sign the drive is dieing even if it is making no sounds?
A hard drive making sounds is not the only sign of a failing hard drive. In some cases, failure will occur at the mainboard... the small printed circuit board attached to the bottom of the drive... which can make the drive just cease to operate. No warning signs, no warning sounds... nothing. Just letting you know that a clicking drive is just one symptom out of many. Not every drive will tell you ahead of time with an audible sound that it is failing.
How can I check to see if the hard drive is the cause?
Among other things, you could run Chkdsk on the drive. It is a utility that is included with Windows. Open up My Computer. Right click on the main drive that you just defragged. Choose Properties. Click the Tools tab. See that Error checking section at the top? Click that button there that says "Check Now..." and put a check mark in both boxes of the little window that pops up. The computer won't be able to check it immediately, it will have to restart and run the program then. Let it do just that. If there are bad sectors and issues with the surface of the drive, it will attempt to repair them, and then provide a report at the end of what it found.
If you are not around when it finishes, you can view this report in the Event Viewer, as a Winlogon entry. Right click on My Computer, choose Manage, expand the Event Viewer, then the Application listing. The Chkdsk log will be listed as having a WinLogin source, and it should be one of the most recent entries in the list.