near miss

American Heritage® Dictionary of the English Language, Fourth Edition

1. n. A narrowly avoided collision involving two or more aircraft, ships, boats, or motor vehicles.
2. n. A missile strike that is extremely close to but not directly on target.
3. n. Something that fails by a very narrow margin: Her campaign for the Senate was a near miss.

Wiktionary

1. n. a miss which was nearly a hit or collision
2. n. by extension a scenario which ends safely but might well have ended in disaster
3. n. by extension an attempt which fails narrowly; a performance which falls just short of a certain benchmark

WordNet 3.0

1. n. an accidental collision that is narrowly avoided