Does an established ssl connection mean a line is really secure
When I visit a website -- say, https://ebanking.pick-your-bank.com -- my browser (fully patched version of Firefox or IE8/9) will display if it finds everything OK with this https connection / website.
How secure am I that everything is actually OK?
Note: This question asks about https security on a public hotspot. I hope to extend the scope to the following points: (not restricted to public networks, also just my private cable-bound connection to my ISP)
- (Local) network?
Edit: I would like to qualify the "security" I ask for: For the scope of this question, let's assume that the servers behind the https connection are in fact secure. That is, the "real" servers haven't been compromised and the data there is decently secured. Let's focus on the connection (also server side) and client/network side of things. (This doesn't imply that one should be blind to the possibility of compromised servers. Let's just not focus on this aspect for this question. Thanks.)