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MarkGreat point, Paul... I know teams often carried on these designs and images for many years. Same holds true for baseball pennants. The Yankees must have used the same portrait of Uncle Sam for over a decade. I have seen dated Yankee Pennants of identical design ranging from the early '50s to early '60s.
Common sense must come into play at some point as well. I always thought those dancing bears looked earlier (and more crude) than circa 1945. I bet that the image was around well before that.
So Frank... the possibility of your Bears pinback being older than mid-forties certainly exists. I would think that late thirties would be about the earliest possibility, but more like early forties (similar to what Paul stated above). Just an educated guess, and perhaps others can shed more light.