(Ok, I know there are inherent problems copying associated elements like notes & dimensions, but in this case, it seemd that the copied elements end up working just fine.)
I am not able to reproduce the problem. I tried to reproduce your scenario by the following methods.
1. Open two files in two sessions of MicroStation. Select a note in one session and copy to clipboard (Edit > Copy). Paste it in the other session (Edit > Paste).
2. Attach file1.dgn containing note into file2.dgn. Use the Copy tool to copy the note from the reference to file2.dgn.
You may be using a different workflow or the problem may be specific to your notes. Can you please send me your dgn file and steps?
In preparing a simple test case (w/o references, extra elements, etc), I was
not able to reporduce the problem - I need to do some more research to
see what's going on.