Another approach might be to eliminate the source of the toppling over. With small prints the origin is the curling of the edges when printing too hot. The raised edges do harden enough from layer to layer to represent a solid piece of resistance for the nozzle. Z hopping can also cause similar raised structures. In this case try lowering the temperature (and speed) for the last layers.
Make sure the first layer is really pressed into the bed. Consider levelling the bed slightly closer than you really should.
Add a bit of glue if you haven't? Oh wait, just noticed you're using a Printrbot and I'm assuming no heated bed? Just tape? If tape make sure it's clean to promote adhesion. Wipe it down with some alcohol to get rid of any fingerprints and/or wax from the surface of the tape.
Try adding the brim in the actual model. This way the brim will be printed differently and possibly hold better. This way you can also make it slightly thicker and stronger.
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