Everything you put on the resume and investigation forms should be verifiable. If they need more contact information they will ask for it before trying to do the investigation. If they find info that doesn't seem to match, they will ask for clarification.
Generally they will give you the opportunity to explain after the investigation, some times they make a mistake in the investigation, or they need help interpreting the info they find.
Example: they contacted the wrong company. I worked for a small company with a similar name, but they called the larger more well known company. Of course they never heard of me. This was easy to fix.
Example: Confusing financial info. A mortgage on an investment property was sold twice before they realized it was an investment property. So the two transactions were reversed, one of which was messed up because my check was already in the mail. The credit report was very confusing, it looked like I had several mortgages I never mentioned. It was easy to explain.
If you embellished your working relationship they could be concerned about what else you embellished. If you accepted payments without paying taxes on the income that could concern some employers.
Bottom line, over explain on the initial forms.