You can, but don't expect to profit unless your electricity is very cheap. Many people mine at home as a hobby with small, quiet devices like the Bitaxe, which work like a lottery ticket for a whole block.
Go deeperFull-size mining machines are loud, run hot, and use as much power as several space heaters, so profit depends almost entirely on your power price. Small home miners use little power and "solo mine," meaning they only get paid if they find a block themselves. The odds are tiny, but a few hobbyists have won entire blocks. Some people run miners in winter as heaters.