Saves to database but says it doesn’t

I just installed WordPress, no problem. I can create a new post & save it, but as soon as I click Publish, I get the “Error establishing a database connection” page. I’ve searched the forum and haven’t seen anything described quite like what’s happening, because it does get published despite the error. Then if I go back and modify the post and click Save, I get the error again, but again the changes are posted correctly to the site. So it’s like everything’s working but I keep getting the error saying it’s not. Any ideas? Is this a WordPress thing or database thing? I also tried deleting everything and starting from scratch with a new database – same thing happens.