Not sure, guess that could happen though. Actually, wouldn't be a bad idea if whether the President was a Democrat or Republican their Vice President was a Libertarian, such a setup might balance things out a little more and discourage partisan politics somewhat.

It's happened before. Lincoln's VP, Johnson, was a democrat. That ended up being a disaster.

In the beginning of this country that was the policy the opposition party Candidate that lost became the VP!!!

Not any more unless the President chooses , when this country started the VP was the runner up in the election

Prior to the ratification of the 12th amendment we did have a president and vice president of different parties. Martin Van Buren was a Federalist his VP was a Republican. That controversy resulted in the passage of the 12th amendment.

As starving artist said it's probably possible but as stick said it would be foolish.

"U.S. Senate: The Senate Elects a Vice President" [senate.gov]