Can you mine bitcoins with an intermittent Internet connection?
Assuming you are mining on a pool, your miner needs to receive and send work (shares) every few seconds, with the exact frequency depending on your hashing speed and the difficulty of the shares.

For example, my ASIC sends and receives shares at least once every 20 seconds or so.

So you need a reliable network connection if you don't want your miner to sit idle. This is the same reason why you normally configure the miner to have fall-back pool servers in case the main pool server goes down.