It has often been cited that NATO promised to not take advantage of Russia's post Cold War weakness. So when NATO proceeded to do just the opposite with countries like Latvia, how Russia might have felt sleighted - hence Crimea - or something like that.
But of course, as indicated in the title, there are people who are arguing that this simply was not the case. I honestly find the counter-claim quite astounding, but the man that says it speaks from a position of authority.
source (circa 19:30)
Problem was, the said "promise" was never put on paper as a formal agreement. Numerous politicians of that era have mentioned it directly or agreed on having "mutual understanding" to that effect. One of top-level western diplomats who claims such understanding is John Matlock.
But there's direct document evidence of informal promise - in phone talk records. German official phone records were de-classified circa 2008.
Here's a quote from Spiegel about phone talk between H.-D. Genscher and E. A. Shevardnadze :
On Feb. 10, 1990, between 4 and 6:30 p.m., Genscher spoke with Shevardnadze. According to the German record of the conversation, which was only recently declassified, Genscher said: "We are aware that NATO membership for a unified Germany raises complicated questions. For us, however, one thing is certain: NATO will not expand to the east."
And because the conversion revolved mainly around East Germany, Genscher added explicitly: "As far as the non-expansion of NATO is concerned, this also applies in general." Shevardnadze replied that he believed "everything the minister (Genscher) said."