Did you finish the edges of the pieces in any way (e.g. FrayStop, zig-zag, etc.)? It could be that the seam allowances just unravelled down to the stitch.
If you still have all your seam allowances intact and persent, it could just be that the stitch you're using is too long. Or that the fabric is just so weak that it can't take the stitching.
Interfacing shouldn't make the thing too stiff, if you only use a little strip of it where the stitches are instead of covering the whole piece with it. And like macabrechaos pointed out, you could try doubling the fabric or using some thinner fabric to double up with (cut the pieces so that the grain of the fabric on one piece is vertical and on the other piece horizontal, so the end result won't stretch in any direction).
I hope that made any sense.