Putin claims that, after the fall of the Berlin Wall, the U.S. promised not to expand NATO beyond East Germany. Does he have a case?

As the situation in Ukraine continues to evolve, many people (myself included) are playing catch-up on learning the history of the region. When talking to a coworker currently living in Ukraine, they sent me this article as a good summary text that is in english. If others have good resources I would love to read them.