The nearest thing to a near miss (and it wasn't near at all) I saw was a slight error in the instruction Heathrow control tower gave to two planes:
Tower: Speedbird XYZ cleared for takeoff
(pilot affirms, starts rolling - it's a 767)
KL ABC: Approaching 09R for crossing
Tower: KL ABC Cleared to cross runway 09R
KL: Cleared to cross runway 09R
(accelerates towards runway)
Tower: Umm, Negative, KL ABC. Do not cross. Cleared to cross after the DEPARTING B767!
(very cool for the situation, the tower controller's voice...)
KL ABC: Cleared to cross after the 767.
Well, near-misses are not exactly frequent, and I imagine such slight and rapidly corrected errors are the closest thing to a near miss that will happen in general.