The line of contact at the moving front of a simple liquid on a substrate can become unstable toward the formation of fingers under the action of an external force, such as a temperature gradient or, a mechanical force. During the dewetting of entangled polymeric liquid thin films, the moving front (rim) of a growing hole can become unstable toward the formation of fingers in the absence of an external driving force. In this paper we show that instabilities develop at the line of contact, leading to the formation of fingers, in films below a critical film thickness, h,. The extent of fingering decreases as the film thickness increases and virtually disappears for h > h,. This critical film thickness increases with increasing molecular weight, approximately as h similar to M-3/7. Our observations are rationalized in terms of a rough scaling argument.