Here is why wire-joints often easily cut-off from the board.
Fig : A. Why wire easily breaks-off. Whole mechanical stress on the delicate copper strands.
B. Fastening the insulating cover of the wire, with the substrate of the board, (or with the insulator of another wire in case of wire-wire connections) using a strong, solid system, reduce the chance of cutoff.
If you could understand why the wires cut-off at joints, you could find-out ways to eliminate this.
previously mentioned solutions by other users, based upon this way of stress removal. The grey layer indicates solder.
However, in test gadgets (not professional) i use narrowly cut (2mm to 1 cm wide) pieces of paper tapes (masking tape) , or tie with threads using the excess holes of the veroboard
By the way, terminus of the wire is twisted before soldering, so that loose strand-ends do-not cause short-circuits.
Image of wire terminus from wikipedia