Someone told me I can't say, someone's words reached someone else nicely.
"His/your words reached me nicely"
Is that true?
You can but it isn't a very common usage. In this context you could replace "nicely" with "just fine" or "effectively":
The likely reason you were told to avoid that usage is because "nicely" more typically refers to the intent or quality of the words:
Switching to "nicely" makes the act of speaking nice and it can cause a little bit of confusion if the reader has to wonder if you misspoke. For the sake of clarity I recommend avoiding this usage but I would not call it incorrect.
Instead of "His/your words reached me nicely," I suggest you recast the sentence to say
You can't really say it because it's meaningless, or at least, its meaning is unclear. 'Nice' once was defined only as 'precise', but modern interpretation of the word includes meanings such as good, fine, pleasant, etc. In other words, your sentence might be saying reached someone else precisely, possibly meaning they were clearly heard, or that the words were interpreted as being pleasant. Either way, it's not something you'd ever hear a native English speaker say, nor see written.
Edit: see Andrew's Comment example of a similar, but contextually correct, use of this word. It's all about the context.