Wouldn't it be smarter to measure password entropy and reject low entropy passwords?

This would allow short passwords using the whole character set to pass, aswell as long passwords only using parts of the character set.

Is the above scheme possible or do implementation details prevent something like this to be done?

Does any site or programm already incorporate such password requirements?

After the famous XKCD strip, there were a few projects started up to deal with exactly this kind of entropy checking. One of these was the ZXCVBN password checker, made by a Dropbox employee.

It is possibly the most thorough password checker of its kind. It checks for patterns, words, and more, adding to (or subtracting from) an entropy score accordingly. It is explained in detail on their blog.
