As designers/developers, I think we're incredibly uneasy with saying "I don't know". In a knowledge-based economy/industry, your knowledge is your currency, so I think there's pressure to appear like you know everything. A combination of being used to presenting a good front to the clients, looking up to rock stars in our industry who can "do it all", and general pride perhaps. The industry also changes so fast, new standards, new technologies, and there's always a fear of falling behind the curve.
I think that's a lot of weight to be carrying around, and we don't always realize it's there.
First, you can't know it all, and know it all well. You can know some things well, even less things really well, but the rest you have to rely on your team, your network or the internet. No man is an island; at best we're an arpeggio that, under the water, is a secret peninsula. Accept it, and focus on what you do well, what you take satisfaction in and what talents people recognize in you.
Second, the rock stars also don't know it all. They know a lot, but they are likely rock stars because they're specialized. They probably don't know some thing that you know quite well. And a lot of what they have done, what *anybody* has done in this industry, has been building on the shoulders of giants. Open source tools, free libraries and frameworks....it's turned coding into an evolutionary ecosystem, where you pick the cream of the crop to get the job done. There's a vanguard out there, sure, they're tackling a lot of these big problems or fiddling with experiements so that we can all benefit, and they should be supported in these endevours, financially or societally. But the large majority of us are doing great work thanks to the great work of other people. This applies more to development than design I suppose, but there's probably something to be said about art influences and such.
In summary: Know what you do well and be satisfied in it, develop it. Know what you can't do well and don't feel bad about it. Be willing to put in the time to learn, have the humility to be taught, the grace to recognize the masters and the generosity to pass it on to others.
(Thanks to Ben Weeks for inspiring this post)