I am developing an XAML Windows Presentation Foundation (WPF) app which is designed to be browser only (an XBAP) and when trying to debug it using a shared database library, I got the above exception when it called SqlConnection.open even though it had an identical string to another working part of the system. I was concerned that it was related to user contexts and session and authentication and other nastys that hide under the covers of a web app but fortunately it was easier than that.
My XBAP application was marked as partial trust which means it is more likely to be loaded without hassle on a strangers browser if you posted an xbap on your company web site for example. This trust level however means amongst other things, you cannot open a connection to a database (fair enough I guess, you could be trying to hack someones pc). If however you mark it as Full Trust (under project properties -> security) it will only be loaded from a trusted location i.e. the local intranet usually unless a user specifically allows it or turns the down the security on their browser but it does mean that you can do more useful stuff like opening connections to databases.
I switched it to full trust since this is for a corporate network and it worked fine (well in fact it came up with an unrelated error but that is another story!).