Make sure that when you're logging in, you're logging in to a page on the site the user will have access to. For example if you sign out from a Site Settings page, the new user logging in may not have permissions to visit that page.
Otherwise though, it sounds like you are doing things correctly. Where are you assigning the user to the SharePoint group? If you'd like to do it from the FBA User Management page, make sure that 'Enable Roles' is unchecked on the FBA Site Configuration page.