It's traditional for winning college basketball teams to "cut down the net" when they reach the Final Four. But just how many people (players, coaches, staff) are allowed to cut down the next?
I.e. can assistant coaches cut down the net?
There isn't a rule stipulating who "can" cut down the net. In terms of numbers, there are typically 12 connections of the net to the rim (which could be cut 24 times if they needed more). Normally, the players on the team (up to 12 per NCAA rules) will cut off a piece of one of the connecting strings as a momento. However, if there are honorary people who helped the team in some way, they usually are also given an opportunity to cut a piece as well (read the following article for such a case).
Typically, the head coach cuts down the last string which brings down the net, but in the case of the 2013 NCAA championship, Louisville's Kevin Ware cut down the last 3 "connections" to bring down the net. (Kevin Ware had a gruesome injury during the Elite 8 and wasn't able to play in the Final Four or Championship game).
Whether assistants or staff do/can, it's really up to the team. This is largely a ceremonial event to celebrate the champions, so it's at their discretion. In my experience, they typically leave it to the players and head coach to do the cutting, but others may get a piece of the net later.
As a matter of history, you can check out this article detailing the origin of the tradition. It wasn't always as ceremonial as it is today.