Are 16 schools too many for a professor to provide recommendation letters to? I plan to apply to approximately 16 schools and am wondering whether that's too many for my recommendation letter providers. 

It's happened before that a professor of my friend refused to provide all letters for him because they're too many. They refused to send all the LoR and cut it from 10+ to about five finally. It would be a disaster if the LoR provider changes his/her mind in the end. Also, I don't want to bother my professor that much.

I believe my professors are nicer than him, but still don't know what's a suitable amount of required schools.

I'm applying for master programs and the 16 selections are range from dream to match to safe, with about five in each category. I failed last year and thus I chose a safer (and might be a bit unnecessary) plan.
As other people have mentioned, the problem with 16 schools is that a professor cannot, either truthfully or operationally provide customized letters to 16 different schools. By "customization" I mean more than changing the name of the school and program. Good letters of rec use professor's familiarity with their field to speak to applicants' specific strengths with regard to the program, or referencing previous students of theirs who attended the program in question. 

In honesty, I wouldn't expect to ask a professor to customize a letter for more than two or three schools, plus a generic one for the others. Maybe four custom letters if you have worked very closely with them.

You could let them know what your top few choices are, then acknowledge a generic letter can be shared among the rest.

I would also reconsider whether 16 is a necessary amount. In my field, 10 programs would be a lot.