When my daughter went through this phase, my go to response was always, "well, then you should try so you get practice until eventually you can. I'll watch you while you try."
If she argued that she shouldn't have to try, I said, "well, the trouble with that is that as your Mom it is my job to teach you things so you can do them on your own, if I just do it for you then I'm not doing my job. Bummer. I guess we can't go anywhere until you give it a shot."
Then I'd watch her try. Often, she was able to do whatever, but it took her longer than it would've taken me which might be why she wanted me to do it. Sometimes she did get stuck somewhere along the line and I could give her useful tips and tricks or help her at the point while encouraging her efforts. "Hey that seemed like a good try, I bet your fingers (or whatever) will just keep getting better at it every time you try."
When she was successful I observed her success, "oh look at that! You got your coat on and you even got one button done! Would you like a little help with some of the other buttons now?"
This method did two things: It showed her I was there and on her team rather than putting us at odds while still insisting on her efforts AND it taught her the value of practice and patience - that it is okay not to be GREAT at it right away.
Now that she is six, she is well-past the "I can't" stage, and when she is facing a frustrating challenge that is proving difficult for her, (hitting a tennis ball in the center of the raquet for example) I often hear her tell herself, "but with practice I'll get it".
This is also what she told my parents when they asked her if she was ready for her first TKD championship the night before. She said, "not really, I'm nervous, but this is a first time so I'll just go and do my best - then next time I won't be so nervous and I'll learn and eventually with practice, I'll get it.
Using this method in combination with Kit Fox's method and making sure to slow it down and understand where your kid is coming from should work wonders.