Your first scenario is one where you make others do their job and the groundwork to establish that there is some kind of connection or speed issue related to the part of the app that does database access.

I'd like to say I take a hard line on that but I don't, because the tools the developers get are often woeful (assuming they're even smart enough to use them) or they're bogged down in process.

What's the solution? Step 1: "I don't know how to troubleshoot IIS. You might need to ask the developers or hire an external resource." Followed by Step 2: "I've taken a baseline. Next time someone asks if there's a problem I can do a comparison to immediately rule the possibility out."

The second scenario you mentioned is different and more about your career path. You wanted to do DBA and they wanted you to do BI. You chose not to and moved on. They might have been rude or dumb about it but don't dwell on these things, the important thing is you're no longer there in a bad situation.