Q:

Is "one too many" too formal or too "proverbial"?

Imagine I had 3 cups of coffee. 2 would have been ok, but 3 is too many. I want to be factual that I had too many, by exactly one. I want to say:

I had one too many cup of coffee.

But I feel like this sentence is a little to "proverbial" in the sense that "one too many" is an idiom. 
According to this answer, it has a "negative connotation" and it's closely related to alcohol consumption.
I don't want to convey that at all. Is my sentence OK, or is there another way to say it?
Also, is there any difference in connotation or meaning with this sentence?

I had one cup of coffee too many.

A:

You are right "one too many" does usually mean there is a bad side effect, feeling sick, needing the toilet urgently, drunk, so full you can't move etc.
This isn't really a formal concept I can not think of a place where I would want to say this formally, as it is a personal internal information.  So every phrase I can think of is casual:

I didn't need that last coffee

is the best alternative I can come up with.

