Darkbreeze :
What happens with air bubbles is, when they burst, you are left with areas that have no thermal paste, therefore heat transfer in those areas is diminished. It's a bad idea, is not a standard procedure and the fact that you're indicating it's ok when every experienced builder I've ever met, talked with or read commentary from all concur that it's a procedure that should not be used. It's just a very bad idea. In a pinch, you could do it for a short amount of time, like a day or two, until you can get replacement paste, but I HIGHLY recommend not doing it.
As I stated, if you've got so much thermal paste that bubbles can form, you're doing it wrong. The vast majority of the heat transfer is from metal-on-metal contact. The coefficient of thermal conductivity of aluminum is about 200 W/m-K. For copper it's about 500 W/m-K.
The coefficient of thermal conductivity for thermal compound is about 0.5 to 5. So it's
much worse than metal-on-metal. Don't be taken in by marking about the coefficient for diamond or silver. When you mix those materials in a suspension, the net coefficient is a harmonic mean, meaning it conducts more like the worse conductor - the filler - than the diamond or silver that's mixed in.
For comparison's sake, the coefficient for air is about 0.024. If you do the math, if a heatsink is sitting on a CPU with 1/3rd of it's surface area metal-on-metal, 1/3rd thermal compound, and 1/3rd an air bubble, 99.3% of the heat transfer is via metal-on-metal, 0.6% via the thermal compound, and 0.07% via air. So when applying thermal compound, it's crucial to maximize metal-on-metal contact.
Which is why if you've got so much paste that a bubble can form (i.e. it's like a layer of mayonnaise on a sandwich), you're doing it wrong. All that paste is probably doing more harm than good. It is in fact possible to run a computer with no thermal compound between the CPU and heatsink with little ill effect, provided there is good metal-on-metal contact (the compression straps help increase metal-on-metal contact).
https://www.youtube.com/watch?v=RAr2wKZ_nes
(And before you go off about it being an "old" cooler CPU, it's a P4 2.4GHz w/ 60 Watt TDP. Modern CPUs probably idle cooler than that one did.)
The reason dirt or hair or dust in the thermal paste is deadly is because it tilts the heatsink relative to the CPU's heat spreader, preventing metal-on-metal contact.