English Translation
I have no patience.
Is this a formal way of saying "我慢できない”?
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It’s a good question, indeed. My short answer is No. They are not always interchangeable. And the difference is not about the degree of formality.
忍耐力がある = 1) be tenacious to get a job done; or 2) be resilient.
我慢できる = 1) be patient not to get upset; 2) be forgivable; or 3) to forgo or manage the situation without something needed.
忍耐 has a connotation of “overcoming a difficulty” while 我慢 is “trying not to concern about a difficulty.” 忍耐 is more goal-oriented.
They are somehow interchangeable, though.
CASE1: Suppose that Tom and Mary are a couple. Mary has a short temper and easily gets upset.
When you say “Tom is 忍耐力がある”, it suggests that Tom carefully listens to Mary’s hysterical complains and tries to understand the root-causes of her emotional swings. The job here is to have Mary calm down. Tom is committed to the job done by coping with such unpleasant complains.
When you say “Tom is 我慢強い”, it describes basically the same characteristics. But 我慢強い also includes the situation where Tom forgives or ignores Mary’s irrational arguments. These tactics don’t solve the problem (i.e. not goal-oriented.)
CASE 2: We say お腹が空いていたが、我慢した (I held back my hunger.) We also say お腹が空いていたが、耐え忍んだ. The former expression better fits when you are supposed to go home and have dinner with your family, so you didn’t have snack on the way home. The latter expression is used when you as a soldier are on the battle field with no food, but keep concentrated, or when the hunger isn’t temporary (e.g. struggling with nation-wide famine.)
The English translation “I have no patience” is vague and can be interpreted as both 忍耐 and 我慢.
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