CAI are a special type of animal. You will see data from people that contradicts each other. Some say it helps and some say you wont see a difference.
I used to work on tuned race cars
and have seen the data on the intake temperatures in different conditions. As the vehicle moves air flow in the engine compartment increases and allows for more fresh air.
For each 1 degree of air temp increase over ambient air temp you will see a percentage loss in HP.
Hence the comments about how winter air at 32 deg F will be better than hot summer air for power because it is cold and dense.
Cliff notes: a CAI will work good at slower speeds by providing consistent cold air to the engine.
An intake that gets its air from under the hood ("short ram") will take in hotter air at slower speeds (i.e. stopped) because the engine heat raises the intake temp of the air under the hood. This increased intake air temperature will cause the engine to pull timing and adjust fuel air ratios to compensate for the hotter air... causing less power to be produced.
Once a vehicle is moving you will not see under hood temps that much over ambient because of all the fresh air coming in.
A CAI will take in the ambient air temp all the time no matter the speed so you will not notice a loss in power at decreased air flow (or vehicle speed)
On my old race car (Subaru WRX STI)
I saw a noticable hp loss for every degree over 85 deg F (the max effective temp for my intercooler heat exchange). So when my under hood temps reached 120 deg at a stop light (only 60 sec stopped), I was almost 50hp down from when I was moving and getting ambient air temps of 85. The air intake temp took over 2 min to cool back down once the vehicle started moving again. I saw under hood temps around 10 deg higher than ambient when moving at moderate to high speed. So a CAI will help when moving, but not much (hence all the data you see about CAI being a myth)
Hope this helps provide some clarification on what a CAI will help with and when.