Here's a little something for Chicago area folks.
Since 2000, there have only been 3 occurrences of a calendar day precip of 1"+ which had less than 1" of snow/sleet and highs in the 30s. Coincidentally, 2 of those days recorded the exact same amount of precip.
Date/high/low/precip/snow amount
4/4/2003: 39/32, 1.22", 0
2/16/2006: 37/22, 1.09", T
3/24/2016: 39/30, 1.09", T