Can I negotiate my checking account fees with my current bank?

Yes. Here are some tips on how to approach this:

See what your fees and charges have been over the past 3 years.
Write down your checking needs, i.e. how many checks you write a month, how many ATM visits, how many deposits, how many times you have overdrawn, how often you go below the minimum balance.
Take this info and do some research into other banks in the area. Compare their rates and fees to your bank.
Go to your bank and ask to speak to a manager. Tell them you want to reduce your banking costs. If they don’t negotiate, bring up their competition. If they don’t want to lose your business they will negotiate. Also ask them other ways to cut costs.
Keep in mind that many banks offer free checking to seniors, students, and the disabled.
Don’t rule out smaller banks as they may be more willing to cut your costs just to get your business.