How realistic is the aviation crisis in the recent Spider-Man movie? In the 2014 film The Amazing Spider-Man 2, there's a sequence in which a villain disables the city's power grid.  All over town, stuff loses power... including the local airport and its air traffic controllers.  This is a problem because there are currently two planes inbound, on courses that will cause them to collide with each other in midair unless the tower can warn them in time!

Luckily, Spider-Man finds a way to defeat the villain and the ATC gets their power back just in time to radio the planes to change course, and they miss each other by an extremely narrow margin.

Putting aside for the moment the question of whether or not a real ATC would have their own independent backup power source, how realistic are the two premises of this crisis?

1) That two airplanes would be inbound towards a major airport on courses that would cause them to both occupy the same position in 3D space at the same time, and

2) That neither pilot would actually notice this until either the tower told them or something suddenly goes CRUNCH!  Do commercial jet liners not have their own radar at the very least?  (Heck, my car has little radar emitters that warn me if something is too close!)
This depends on the aircraft type, the flight rules and weather conditions as contributing factors.

Larger aircraft or medium sized aircraft operating under general aviation or commercial aviation will in many cases be equipped with a technology called Traffic Collision Avoidance System (TCAS) (See: How does TCAS work?), which can instruct both aircraft to take evasive action if a collision is imminent. This is especially important in Instrument Meteorological Conditions, where pilots do not rely on outside visibility, but on instrument flight only operating under Instrument Flight Rules.

In Visual Meteorological Conditions, aircraft would likely be able to see each other and take evasive manueouvers without ATC or TCAS having to warn them, e.g. this is the case for smaller aircraft operating under Visual Flight Rules.

Aircraft receive clearances and instructions to occupy a certain flight level or altitude, sometimes coupled with clearances for approaches or arrival/departure procedures. If pilots do not receive further instructions, they will start to hold over their last clearance limit and commence lost radio/comms procedures, which include holding for a given amount of time and then beginning the descent towards the airport on defined and published approaches. If the power outage is long enough, the airspace is quite busy and all other frequencies are also unavailable, one of the being the distress or guard frequency, there is a probability that two aircraft on the same level will reach their clearance limit at some point and enter a hold at the same flight level, which could result in a collision.

The scenario however assumes that really all failsafes and contingency procedures have failed.

See also ths related question: What happens when an airport shuts down?