Google Drive won't let me download my files

Try logging out of all other users in your web browser or use Incognito Mode if your browser has it (which has the effect of doing this). Kudos to Ahmad for saving me with this answer today!

As he clarified (and glenviewjeff and Boynux both suggested in the comments to the OP), either of these approaches seem to help because of a bug with multiple user sign-in. It doesn't even matter if all the accounts signed in are granted permission to download the file; it's something about having more than one account signed in.

I had this happen today with .xlsx files in both Chrome and IE11. In IE11 I saw the same behavior as the OP described here:

"when I pressed the download button, Drive opened another tab in my browser with an Error 403 "Forbidden" on it. ... They also gave me an Error 403."

In Chrome the behavior was a little different (it opened a download window prompting me to save a file with a long string of random numbers and letters and no file extension; clicking save returned a "Failed - Forbidden" message in the download tray at the bottom), but the result is the same as the OP's subject line: Google Drive won't let me download my files.

This works for me: Right click on the file, then click on Create shareable link and use that link.