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There was no such promise. The promise was to guarantee the very borders Putin is violating.
The NATO grew because the peoples of East-central Europe are afraid of the Russian Federation. They are not afraid of Germany, England or the US. The experience of the post-WWII Europe taught them whom to fear and whom not to fear; they learned their lesson.
http://www.spiegel.de/international/world/nato-s-eastward-expansion-did-the-west-break-its-promise-to-moscow-a-663315.html
"Russian President Dmitry Medvedev has accused the West of breaking promises made after the fall of the Iron Curtain, saying that NATO's expansion into Eastern Europe violated commitments made during the negotiations over German reunification. Newly discovered documents from Western archives support the Russian position."
http://www.globalresearch.ca/russia-or-america-who-is-the-real-aggressor/5372882?print=1
"While Russia has been adhering to international law and showing complete lack of aggression over the last 14 years, the US and NATO have been doing the opposite. US alone has intervened in the following countries: Sierra Leone, Nigeria, Yemen, Afghanistan, Philippines, Côte d’Ivoire, Iraq, Georgia, Haiti, Djibouti, Kenya, Ethiopia, Eritrea, Pakistan, Lebanon, Somalia, Libya, Uganda, Jordan, Chad, Mali, Turkey. NATO, meanwhile, has been involved in the Bosnia and Herzegovina intervention, Kosovo intervention, the Afghanistan War and most recently the Libyan intervention; with the latter two being complete disasters that have the left the countries in shambles and anarchy.
In addition, NATO has broken its promise to Russia. Mikhail Gorbachev reportedly agreed to allow German reunification within NATO after being promised that NATO would not expand “one inch to the east.” Instead, in 1999, Poland, Hungary, and the Czech Republic joined the organization, despite NATO claiming that it had no plans to expand after the end of the Cold War. Another expansion came with the accession of seven Central and Eastern European countries: Estonia, Latvia, Lithuania, Slovenia, Slovakia, Bulgaria, and Romania. These nations were first invited to start talks of membership during the 2002 Prague summit, and joined NATO on 29 March 2004, shortly before the 2004 Istanbul summit. Most recently, Albania and Croatia joined on 1 April 2009, shortly before the 2009 Strasbourg–Kehl summit. Future expansion is currently a topic of debate in many countries. Cyprus and Macedonia are stalled from accession by, respectively, Turkey and Greece, pending the resolution of disputes between them. Other countries which have a stated goal of eventually joining include Bosnia and Herzegovina, Montenegro, and Georgia."