suffix, i-adjective
cannot completely ...; too much to ...
Attaches to the -masu stem of a verb to express that the action cannot be fully completed because the amount or degree is excessive. Often used with verbs like 食べる, 数える, 持つ, etc.
See also: 切れる
こんなにたくさんの料理は一人では食べきれない。
I can't possibly eat all this food by myself.
星が多すぎて数えきれない。
There are too many stars to count.
切れる is the positive potential form meaning 'can completely do', while 切れない is its negative counterpart meaning 'cannot completely do'.
Negative potential form of 切る (to cut), used as a suffix meaning 'to do completely'. The negative 切れない thus means 'cannot do completely'.