Firstly, most of the ethnic minority groups in the Soviet Union did not achieve independence. USSR had 128 (or so) officially recognized ethnic groups within it.

Secondly, Chechnya actually proclaimed independence during 1991 as Chechen National Congress disbanded previously existing Chechen-Ingush Autonomous Republic and its government in September 1991 and held their own elections in October.

Russian president Boris Yeltsin refused to recognize the elections and tried to impose martial law on the Republic, but Soviet military bases and law enforcement offices were blockaded by militias raised by Chechen National Congress and situation ended in impasse. No large scale fighting broke out, but in following months all Soviet (now Russian) military personnel vacated the Republican territory, leaving most of the equipment behind.

Actual Russian military intervention happened three years later after independent Chechnya descended into internal civil war.