Is this problem just an application of traveling salesman? If not is it some other already "solved" problem? Description of the problem in question:

Say I have a complete graph with positive weighted edges. 1 vertex is specified as the "end". A subset of the other vertices are designated as "start" vertices.

I want to find paths from each start vertex to the end vertex with the following conditions:


As input, each path is given a maximum number of vertices it can visit. ie a path with a max of 5 vertices can only visit 5 other vertices before going straight to the end.
Each vertex can only be visited by one path
Each path must visit its max number of vertices, as long as there are unvisited vertices still available.
The total combined cost of the paths should be minimized


Basically, I want to find a path from each start vertex to end vertex that hits as many other vertices as possible, limited by each paths max and that each vertex can only be visited once. So it's possible that not all vertices will be hit or that a path won't visit its max number of vertices.

I want to know if and how this problem breaks down into TSP, or some other problem. In general, I want to figure out how to solve this problem but I'm just looking for a place to start.

Any insight would be greatly appreciated.

Thanks a lot
This problem comes down to a transportation problem (actually the Wikipedia article is not very good), although it is not exactly the problem you are stating. As David Richerby stated in the comments, your problem is formulated contradictory, therefore I would suggest a reformulation, where you fix the number of vertices that each path visits. But while reading yourself into the capacitive transport problem, you will find out yourself what changes fit best for your task.
Solving these problems can indeed, as stated by Raphael, be solved via min-cost max-flow algorithms. 
When you decided on a new version I am happy to extend my answer to fit your needs.