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I ran across 不知不觉 the other day. If I use MDBG to look up c:不*不* I see there are many idioms with that format. Can I use it with any verbs (?) or is it only part of fixed expressions?

Is 不 A 不 B a grammatical structure like 又 A 又 B?

They are different. Most [不 A 不 B] terms are either idiom or four characters idiomatic phrase that you have to memorize individually.

The A and B in [又 A 又 B] can be a single character word, a compound word or even an entire phrase. e.g. 又高又瘦，又高又瘦削，又想高大又想瘦削

Below are the common [不 A 不 B] terms:

[A complement B ]

不慌不忙: (not panic, not hurry) stay calm

不聞不問: (not listening, not inquire) ignore

不知不覺: (not know, not feel) not notice

不經不覺: (not experience, not feel) not notice (of time passing)

不瞅不睬: (not looking, not pay attention) ignore

不明不白: (not understandable, not clear) unclear

不倫不類: (not any type or kind) ridiculous

不慍不火: (not angry, not enrage) calm; mild

不痛不癢: (not hurt, not itchy) insignificant attack/ lost

不飲不食: (not eat, not drink) not eat nor drink

不眠不休: (not sleep, not rest) tirelessly; relentlessly, persistently

不忠不孝: disloyal, not filial (no virtue)

不仁不義: unkind, unjust (no honor)

不過不失: (not considered a wrong doing, not considered a mistake) average/ mediocre

不折不扣: (no discount, no rebate) truly; definitely

不離不棄: (not leaving, not abandon) never abandon

不盡不實: (not detailed, not true) not telling the truth

不驕不躁: (not prideful, not irritated) stay calm and focused

不屈不撓: (not bending, not yielding) Never give up

不三不四: (not this, not that) dubious

不依不饒: (not agree with, not forgive) pestering/ not let thing go

[A contradict B ]

不卑不亢: (not too humble, not too excited) proper demeanor

不偏不倚: (not lend away , not lean onto) right on/ Objective (opinion)

不緩不迭: (not slow, not swift) steady normal pace

不男不女: (not a man, nor a woman) sex unknown

不上不下: (not up, not down) struct in the middle

不大不小: (not big, not small) the right size

不冷不熱: (not hot not cold) mild reception; right temperature

不高不低; (not high, not low) middle position (of rank or level)

不胖不瘦: (not fat, not thin) average body type

不咸不淡: (not salty not bland) unremarkable taste/ mediocre

[B until/ unless A]

不死不休: not stopping until one is dead

不吐不快: not happy until saying (something)

不時不食: do not eat unless it is in-season

不見不散: not leaving until we meet

I would update this list when I think up more.

why not? see bkrs: II cons.
1) 不 A 不 B (where A B are paired synonyms): not A B
不慌不忙 unhurried
2) 不 A 不 B (where A B are antonymous stative verbs): neither A nor B
不大不小 neither (too) big nor (too) small
3) 不 A 不 B (where A B are antonymous verbs): no B without A
不破不立 no constructing without destroying
cf. restrictions for 又 。。。又, see e.g. https://resources.allsetlearning.com/chinese/grammar/Expressing_%22both_A_and_B%22_with_%22you%22 bkrs:又.... 又… linking antonyms, often translated as neither... nor, 又不好, 又不坏 , 又不哭, 又不笑,
[又]不是牛又不是马

No, not with any verb, although you would see lots of idioms follow that pattern. You need to learn them separately.