What kinds of problems do you encounter? Is it missing parts, wrong part numbers, wrong quantities? I think what will help a lot is to try to include everything in the assembly before generating the BOM. Especially hardware, and then browse through the assembly making parts transparent so you can verify that screws are the correct length.
Are the engineers generating their BOM that goes to costing from SW? If not then I'd say that they should. I keep an excel BOM for a new product that is constantly changing as the product goes through constant changes in R&D. But before I give anyone a final BOM when it is released to production I generate a brand new BOM in SW and export that to excel, then go back through it and fill out other info like mfr, mfr part number, etc. If you do this, then you should never be missing items in the BOM, unless the engineers are deleting rows. If they are then that is a concern of its own. You may be missing items that are in the BOM but don't have a SW model, like paint or epoxy or loctite. But those things shouldn't be too catastrophic if they aren't discovered until later.