iCloud handles only its own email; it cannot access other email services. Therefore thare are only two possible explanations:
1. Your iPhone is collecting mail from the texasbb account as an independent account and it is appearing in the Mail app alongside the messages from iCloud.
2. You do in fact have forwarding set up on texasbb to your iCloud account.
If you check you iCloud email at http://icloud.com, does the texasbb mail appear there? If so, 2, if not, 1.
Thanks Roger, But I'm still stuck. The texasbb mail does show up in my icloud account when i'm logged into http://icloud.com. A call into Texasbb.com told me they don't offer mail forwarding thru their servers, so something else is forwarding it. No way icloud could fetch mail the way gmail does?
There is another possibility. Were you originally a MobileMe subscriber, and if so did you have MobileMe set up to fetch your Texasbb mail? Some people have found that when they migrated to iCloud the fetching continues (even though it's not a facility which iCloud provides) and because the facility isn't available there is no way to turn it off. If this is your situation you may be able to persuade iCloud Support to turn it off - I don't know how likely it is that they will or can. Go to https://getsupport.apple.com ; click 'See all products and services', then 'More Products and Services, then 'iCloud' and proceed from there.