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gbj_tester
This is the same problem as the seven bridges of Königsberg.
The simple explanation is:
Your path has one start point and one end point. Aside from the starting line and end line which arrive at these points, at every other corner/vertex you must both enter and exit, which requires an even number of edges/lines. For each of the 4 vertexes/corners, there are odd number of edges or connecting lines. Since there can only be 2 (or 0) odd numbered vertexes (for start and end), you are guaranteed to eventually land on a vertex which has no exit. So it can't be done.
If you add a second line across the top or bottom, you have made the number of vertexes even on the two connected corners, which makes it work