David,
ArcGIS cannot read all of the details of a dgn file, unfortunately. Some important aspects are left out when it converts over, leaving lots of issues.
I know that FME software uses some sort of converting tool that seamlessly takes dgn data and imports it into GIS (in whatever format), but of course, that comes with a subscription cost.
I have not seen the attribute information being really different from one dgn to another when it should be the same.
What I always see are these attributes from a dgn:
FID. Shape, Entity, Level, Layer, LyrFrzn, LyrOn, Color, Linetype, Elevation, LineWt and RefName.
The "level" in microstation seems to coincide with "layer" in GIS. And mainly only the color, linetype, and linewt are the only things that change for me. The others kind of stay the same or are completely irrelevant.
Can you show some specific examples of what you're seeing and how they're different?