I'm currently reading a book called The Prophets of War: Lockheed Martin and the Making of the Military-Industrial Complex and in it William D. Hartung makes the claim, citing Leon V. Sigal's book Hang Separately: Cooperative Securtiy between the United States and Russia, 1985-1944, that the US promised the Soviet Union that they would not extend NATO membership to Eastern Europe.
Different sources argue either way. This SPIEGEL article and this journal article in the Washington Quarterly [PDF warning], for example.
According to Gorbachev, Baker did make this pledge; according to Baker, he was never referring to anything other than Germany. Within Baker's personal handwritten notes of his meeting with Gorbachev he put exclamation points next to this statement regarding the expansion of NATO:
"End result: Unified Ger. anchored in a changed (polit.) NATO--whose juris. would not move eastward"
The bold represents the words and phrases that Baker put stars by.
A few days later, on February 9th, Gorbachev repeated to Baker that he couldn't accept a NATO that would expand eastward. Baker said that "we agree with that."
Mary Sarotte analyzes the diplomacy between Baker and Gorbachev particularly well in her book 1989: The Struggle to Create post-Cold War Europe. You'd probably get more from me copy and pasting the main part of what Sarotte says on the exchange between Baker and Gorbachev rather than me trying to summarize it. It begins on p.110:
[The agreement between Baker and Gorbachev] formed the nucleus of the controversy that remains unresolved to this day. Unwisely, Gorbachev let the meeting end without securing this agreement in any kind of written form. Emerging from a political culture in which the word of a leader overruled the law, hoping that he could still find a way to disband both military alliances [NATO and the Warsaw Pact] entirely, and hesitating to agree to his end of the bargain (a unified Germany), Gorbachev did not try to resolve the matter there in writing. In the future, once NATO started expanding, he would therefore leave the Soviet Union's successors empty-handed when they protested against NATO enlargement. Later, Russian presidents would assert that this meeting had given them assurances that NATO would not expand. The United States would remember this meeting differently: as one in a number of conversations and negotiations limited solely to Germany, and until the final documents were signed, changeable.
To put this meeting into greater context, you may want to see if you can perhaps read all of chapter 3 in Sarotte's book, of which the above quote is a part of. Or you can just buy the book and read the whole thing; it's an absolutely fascinating read.
Edit: Clarification
This comment may not belong here (it doesn't answer OP's question, but rather tries to take the discussion to another level), but I will try.
I'm always hearing from pro-Russia folk like Stephen Cohen at The Nation that because of this "broken promise", Russia feels like it has been treated as a defeated nation. People like this usually go on to argue that Russia deserves its own sphere of influence and is right to object to NATO's eastward "encroachment".
To this I say: beyond the real politik concern of "poking the bear", why should we care? If former Warsaw Pact nations want to join NATO, why should Russia be given sympathy? IMO, too bad; so sad. I guess a half a century of Soviet control wasn't enough to engender fuzzy feelings for the Russian state. Does Russia's sphere of influence mean the west should let it dictate to former client states?
Finally, with respect to Russia feeling "threatened by NATO", does anyone, Russians included, honestly believe that NATO has any desire to invade? I just can't believe that one of the Baltic states is just going to wake up one morning and decide to attack.