Part III at the top says: “You must complete this part if you (a) had over $1,500 of taxable interest or ordinary dividends; (b) had a foreign account; or (c) received a distribution from, or were a grantor of, or a transferor to, a foreign trust.”
This type of language should be read as:
You must complete this part if you:
(a) had over $1,500 of taxable interest or ordinary dividends; or
(b) had a foreign account; or
(c) received a distribution from, or were a grantor of, or a transferor to, a foreign trust.
It should not be read as an “and” between (a) and (b). If you had to complete Part III only where you had a foreign account, then no one would ever answer “no” on question 7a. This wouldn’t make any sense.
If your income is below the filing threshold for a Form 1040, then I don’t think you need to file at all, even if you have a foreign bank account. On the other hand, if you have sufficient income to be required to file Form 1040, I think you have to fill out Schedule B, Part III, regardless of whether you have any dividends or interest income.