I found the same situation, living in China for quite some time, and unlike some other people who have answered, I understand exactly what you're asking. It was quite annoying to try to learn new words when the native speaker just tells you the meaning of 3 characters together and doesn't know or can't explain each character's meaning. I think the answer is related to how derivations work in English. Keep in mind that to us, the word duibuqi seems to be 3 words together because technically it is, but to Chinese people who aren't language experts, they will just think of it as one word, essentially.
There are some parallels in English. Think about it in the reverse. Imagine you are Chinese and you are asking an English speaker this question:
You say 'sorry' to mean duibuqi. What is 'sor' and what is 'ry'?
Can you answer that off the top of your head? Most English speakers incorrectly assume that prefixes and suffixes and word derivations aren't "words." They are words. Or at least they used to be words.
From an online derivation dictionary, this is how we have the word sorry:
Old English sarig "distressed, grieved, full of sorrow" (not found in the physical sense of "sore"), from Proto-Germanic *sairiga- "painful" (cf. Old Saxon serag, Middle Dutch seerigh "sore; sad, sorry," Dutch zeerig "sore, full of sores," Old High German serag, Swedish sårig "sore, full of sores"), from *sairaz "pain" (physical and mental); related to *saira- "suffering, sick, ill" (see sore (adj.)). Meaning "wretched, worthless, poor" first recorded mid-13c. Spelling shift from -a- to -o- by influence of sorrow. Apologetic sense (short for I'm sorry) is attested from 1834; phrase sorry about that popularized 1960s by U.S. TV show "Get Smart." Related: Sorrily; sorriness.
So it appears the answer to what is 'sor' and what is 'ry' would be:
"Sor" means grieved or distressed
"ry" (from online dictionary) means "indicating a state or condition."
Therefore while you may be asked by a Chinese speaker "What is sor and what is ry?" and you will feel like saying "Nothing, it's just one word," in this kind of context you would be wrong. They are two words indeed, we just don't usually use them separately. Sound familiar?
Disclaimer: I'm not a native Chinese speaker but the above answer does seem to be the most rational, to me.