Not being able to blink all the way?!

Not being able to blink all the way?

I wanted to get contacts but the eye doctor said that I can't because I don't blink fully and that it would cause irritation.

Is this possible? If so how can I fix it?

I understand what your eye doctor is speaking of it is called a partial blink. It has nothing to do with the rate you are blinking it is when you blink your eyelids are not completely coming together. This will mean that you will not be able to wear rigid lenses, unless your blink is corrected or the rigid lens has a lid attachment design. Soft lenses are a possibility but you may have some difficulties with those as well.
Incomplete blinking often can cause soft lenses to become dry and their surface can become somewhat hydrophobic and coated with a deposit. The rate of deposition may increase and the quality of vision may be adversely affected.
People with poor blinking habits should routinely use an enzymatic cleaner at least once every 6 days to minimize protein build up on the front surface. Also use of rewetting drops will help keep your contacts hydrated.
To fix the partial blink you either have to address the underlying cause of it or try blink exercises to correct a faulty blink. If your doctor is still hesitant then you still may be able to wear contacts for social events or "part time".
I think you should ask your eye doctor if you have any other options.... If they don't give you any you may want to get a second opinion.
Hope that helps :)