It's a Team Sport
For both employees and managers, it's important to realize that it's a team sport. If the employee can (and does) teach and work competently with A, do pretty well, most of the time with B and is only struggling with C, then it's important for everyone to realize that this is pretty much normal. The employee should work on striking a balance between helping others with A and learning C - that gives the whole team a feeling of shared contribution, and it helps tremendously with confidence when struggling with learning C.
Case in point - my The Workplace participation skyrocketed a while back, when I'd found a new job - feeling incompetent starting out in a new field, I was thrilled to be able to provide advice on what I'd just done pretty competently - find a new job!
So, also - realize a "team" is more than just the people you work with - and use forums like these as a place to both find and offer answers.
A good manager will realize this, too, and work out a plan where the idea of a "good" employee is one who can both ask and answer questions - who is great at some things and still learning others. So a person isn't rated on never failing, but on both his ability to try new things (and fail sometimes) and help others as they do the same.
There is a Learning Curve
Ponder previous learning curves and realize that the current one will have some striking similarities. My favorite is the Four Stages of Competence. Having a sense that there is a way to go through a learning curve, and that you can measure your own progress from week to week, and set mini-goals for yourself can be extremely encouraging. Some of my mini-goals have been:
Know what all the words mean when someone answers my question?
Be able to rephrase an answer in my own words
Understand why what I did didn't work, and why the actual fix did.
Little things like that lead to the bigger things like having a mental model for the technology, or being able to work out a simple problem unattended.
Often, when school was the only place a person's learned that first learning curve in the working world is hellish - it's a different process, and the lack of feedback will make you crazy.
Last Resort - ask for Feedback
I don't mean to imply that you shouldn't ask for feedback ever. But getting a hard new assignment and asking "how am I doing?" one week in is too soon. Asking 6 months later is too late. As a rule of thumb, give yourself 3 failures and 3 successes and then check in with a senior member of the staff or your boss. That works well for the technique - 3 up/3 down feedback giving - you and the reviewer each say 3 things you did well, 3 you could do better. I haven't tried this yet, but I like the idea quite a bit, as it's takes the review away from a judgement call on "you were good/bad" and into the point of view that everyone has things they can work on and things they do well.
Change is Good
I call it that, because "grow or die" sounded too harsh. Realize that if your job currently seems impossible, it may just be that it's because you were doing the last one so well. Being trusted to take on hard stuff is a mark of distinction - being able to do so is what distinguishes you further.
If all you do is all you've ever done, then your career in technology will be short-lived. Every area of technology is changing and innovating too fast for anyone to stay still and not learn new things. So, my last advice would be that learning how to learn is never going to become obsolete!