Teething

Is it possible for a one month old to be teething already? Our son normally only cries if he's hungry or needs his diaper changed. Other than that, he gets fussy if he's fighting sleep, wants his binky/tucky/pacifier, or if we're in the car and it's stopped for about 3+ mins if he's awake.
This week we bumped the bottle up to 5oz and thinking we need to bump it up to 6oz already because in 1-1.5 hours, he's hungry. He has quite a few wet diapers and BM's every day.
Today, since about 6-7pm, he's been crying almost nonstop. Hubby got him to sleep about 10:30 then I rocked him to sleep again a little after 11:30..its 11:52 and he's crying again.