Something that you will eventually get and continue to get once you become a Cop. Getting the "thin blue weenie" may include such fine events as: getting fucked with a shitty report calls because your P.O.S. partner is the sergeants pet, getting in trouble for some small petty shit off duty, getting called in from vacation for some unnecessary in-service training.

“Thin Blue” as an allusion to the “Thin Blue Line” police officers so proudly walk. The "weenie" represents getting fucked over.

when you're hungover as shit after a spending a night drunk as a skunk