I can confirm x2jroy's fix comment. However, this doesn't happen on every Droid X.
I have a Droid X and do not have this problem. My girlfriend's Droid X started failing at downloads (the same way x2jroy described in the original post).
From Home screen: Press the physical "menu" button on the phone and select Settings from the popup menu.
Navigate to Applications >> Manage Applications >> Market (under the Downloads tab).
If the version is 2.12 AND you are having this problem exactly as described (download icon appears for 2 seconds in the droid menu bar then disappears without warning or error), then the fix is to hit the Uninstall Updates button (next to Force Stop button at the top). This will only remove updates to Market, it will not uninstall the whole app. This does fix the issue of being unable to download.
Using the Uninstall Updates button alone will cause you to have to Agree to the Market Terms of Service; that's how you'll know your next download should work.
Oddly, if you go back to market after downloading anything successfully, the version of Market is back to 2.12, and downloads will once again cease to function until you repeat the fix above of uninstalling Market updates. Then you can download one more app. Rinse, repeat, rinse, repeat.
I don't know if using the "Update All" button will work after this fix, or if it will crap out after the first completed install.
Very annoying.
