Is it possible for computing power to increase quick enough to break mining retargeting?
The difficulty of the mining algorithm is solely reliant on the time it takes to mine 2016 blocks and the previous difficulty. (See: How is difficulty calculated?). 

As the mining algorithm retargets exponentially (up to a factor of 4) every 2016 blocks, it is impossible for technological progress to outpace difficulty increase in the long run. Even if the mining power were to double every day (which is insane!), the block interval would only go down to about every two minutes in average, i.e. approximately five times faster.

On another answer I've provided a sample calculation for an overnight hash power increase by factor 21, which you might find interesting. The difficulty would catch up in 14 days.