I'm no expert here, but, in my experience - if your refrigerant level gets too low, it won't kick in your compressor/clutch when you hit the snowflake (AC) button. I've been able to infuse a can of refrigerant and, during the process, the compressor clutch kicks in and things start working.

But, as I'm sure you know - there could be a host of other issues; leak in the system/bad compressor/etc.