Consider the impact wrench. This is a honking big tool that demands a lot of compressed air flow and pressure.
All of us backyard mechanics dream of using one of these, hearing that wonderful brraaaaapp as the wrench starts to turn a too-tight lug not off.
Yet, if you have a .5 HP compressor with a 2 gallon tank on board, you will run the wrench for only a second or two before you are out of air.
When buying an air tool read the specs. Somewhere it will tell you that this air tool requires so many CFM of compressed air at so much PSI.
A typical air compressor will deliver 3-4 CFM of compressed air at around 90 PSI for every HP of electric motor size.
This means that if your air tool needs 10 CFM of compressed air at 90 PSI to run properly, the math tells you that you will need a 2.5 HP air compressor to provide the compressed air for just that one air tool.
Yes, having a large air tank will help you run a high-demand air tool with a smaller compressor. With a smaller compressor bigger-tank combo, this means that your air compressor is going to work very hard to try and keep that tank full as you use your air tool, and you will still exceed the air supply available, and have to stop work and wait for the compressor to catch up.