Error 'Insufficient Privileges' when an Opportunity or Case is created

This may occur if the user is granted access to the related account only through a sharing rule, which specifies "Private" access to related Opportunities or Cases.

We may check the user's access to the account by using the "Sharing" button on the account detail page.  We can resolve the issue by updating the appropriate sharing rule to include Read/Write access to the related Opportunities/Cases.  Sharing rules may be viewed and modified by following this clickpath:
 

In Classic UI:

1. Go to 'Setup'
2. Under 'Security Controls', click 'Sharing Settings'

In Lightning UI:

1. Go to 'Setup' (Gear Icon)
2. Under 'Security', click 'Sharing Settings'

See also
Sharing Rules
Resolving Insufficient Privileges Errors