Only sort of.
An air compressor has two main components: the pump which fills the tank.
For 'sustained' use, you need a pump which can pump enough air to run the tool(s).
For 'intermittent' use, you can sort of rely on the compressed air that is stored in the tank.
Think of the air tank as a rechargeable battery. The bigger the battery, the more you can use it before you have to recharge. But you will have to recharge it sooner or later, and then the time it takes for the battery to get 'refilled' depends on the charger. (leaving out some of the 'finer' details, etc, etc, etc).
So although you can (within limits, see the rest of this paragraph) add more air tank storage/capacity, that means that the pump will have to run longer to refill all that capacity. And most compressor pumps are made/designed to only run a certain percentage of the time (duty cycle), and then they need to stop and cool off. Most units have the pump size and duty-cycle roughly matched to the tank size and general 'usage profile'.
So although a little compressor pump that might come with a little 1 gallon tank could (theoretically) fill a big 80 gallon air tank, the little compressor pump will definitely suffer from a reduced (greatly reduced) lifetime and will likely self-destruct pretty soon if it ran long enough to try and fill the big 80 gallon air tank.
For air compressors, you generally want to size it based on the CFM the compressor can deliver and the CFM of the tool(s) you want to run. And then get 2x the compressor CFM as the tools need (ex: tool needs 5 CFM, so try and get a compressor that can supply 10 CFM). Note that some tools are 'rated' on an intermittent basis for the CFM demand, and the sustained CFM can easily be 4x as high or higher (ex: an air die grinder may be 'rated' at 5 cfm air usage, but that is based on an intermittent air usage of just 15 seconds use in 1 minute elapsed time, so if you will be using that grinder on a sustained/continuous basis you would need an air compressor that could supply 20+ cfm).
A little 1 gallon air compressor can run most air nailers OK, but you may end up waiting for the tank to refill before you can fire the next 'shot' if you have a bigger nailer (like a framing nailer). YMMV.
In general, get the biggest air compressor you can afford and fit.
Almost no one ever says "Gee, I wish I had gotten the smaller air compressor."

(larger industrial screw air compressors exempted there)