|Leonard Dubkin Park|
According to Tom Skilling the only completely snow free months of the year in Chicago history are July and August. The biggest April snowstorm in Chicago (according to the Trib) happened on April 2nd-3rd 1975 with a total of ten inches.
Eight times significant snow has fallen in May, the highest total was 2.2 inches May 1-2 1940.
Consulting the National Weather Service site we see that April snowfall isn't so uncommon. NWS site states that the normal snowfall total for Chicago is 1.2 inches and the highest accumulation was 13.6 inches in 1938.
Chuckman's Chicago nostalgia site has a great April snow photo (woman writing on her windshield).