If the amount in your account is larger then the check, I would expect the bank to cash it, but you may also find there is a hold on that amount until the check clears.
I've found banks have always made a difference between flat out cashing a check and a check where a portion is also deposited. Going in with a $3,000. check and depositing $1,000. taking $2,000. in cash, I'd expect to go through no problems. Again there would have to be enough on account to cover, and may be put on hold, ask.
Bank of America doesn't care if the check is from their account holder, if you don't have an account with them they charge a fee or require you open an account. Three months ago watched a guy go through this trying to cash his lawn service pay check, but hey this is Florida, things are different here. The funny part was they had cashed the check, he had the money in hand, then they realized they expected him to pay the fee. He was saying the transaction was finished, they disagreed, then it was my turn and lost the rest of the story.
With so many scams out there with fake cashier's checks that first appear to clear and later bounce, you may find the bank taking some measures to protect themselves and their customers.
Tip: If you have a garbage disposal the day will come when it is going to leak ruining your cabinets and making a big mess. Keep a bucket under it, not high sided enough to have the disposal sitting in water as there is electric, and so you can remove the pail without having to tip it. Added to the list of, wish I had done that.