The vast majority of textbook exercises are worded essentially in the format:
This assertion is (true/false). Prove this or find a counterexample.
This, of course, is not how mathematics is done. In the "real mathematical world", the questions you pose have unknown answers, and thus there isn't such an obvious structure to solving the problem. While it takes some creativity and frustration to come up with a proof that you know exists (only a Google away!), it seems vastly harder to try and actually solve a problem where the definitive answer is unknown.
Since I know many researchers on this site work on such problems every day, I must ask: how does one go about getting started on a problem like this? Do you try to write a positive proof and see where it fails? search for counterexamples by intuition alone?
I'm by no means someone who does research, but I don't see this huge difference.
If you find the question on a textbook you have the following theorem at your disposal : "There exists a solution that employs only theorems I've already seen and it is reasonably short and relatively easy to find "
This is an important factor because it keeps you determined and prevents you from giving up too fast, and maybe it will boost your confidence, but if you keep focused it won't be too different.
(Of course, George Dantzig's story is an exception ;-) )