I've read a few responses, and the majority seem to answer with a no.
I'm going to answer with a yes, but then I need to qualify it.
The doctrine of total depravity is a doctrine held by both Calvinists and Classical Arminians. There is an important reason why it is critical, and this takes me back to the controversy between Luther and Erasmus. If memory serves, Luther congratulated Erasmus on locating the critical point of conflict. And it revolves around the self-sufficiency of the will vs the bondage of the will (as Luther put it). The doctrine of total depravity gets to the very heart of the issue, which is this. For true salvation to occur, and for one to have genuine, empty-handed faith, self-sufficiency must be destroyed, and the doctrine of depravity is the nuke.
The critical issue is this. One may affirm the need for faith, but then "faith" is couched in terms of self-accomplishment. Another may affirm the need for faith, but then "faith" is couched in terms of utter self ruin, and thusly faith is completely in Christ and His work and merit with no divided allegiance. The eternal danger is that one may use religious language, Christian language, but in the end their "faith" is a self-salvation because of the inherent self-sufficient nature of the will they describe. Equivocation here (over the very meaning and essence of "faith") may be eternally damning. All of this connects to the doctrine of depravity and its connection to the will.