The Plot Thickens is an achievement/trophy in L.A. Noire. It is earned by finding and solving an inspection puzzle.
While it can be achieved on any case involving an inspection puzzle, the earliest and among the easiest opportunities to obtain the achievement is the very first Traffic case The Driver's Seat.
During the inspection of the crime scene at the P.E. Freight Depot, make sure to examine the bloody InstaHeat pipe to the left of the vehicle between the forked train tracks, at the evidence B marker. Later when inspecting the Black Residence, pick up and examine both the InstaHeat receipt and flyer (showing the assembly diagram) on the kitchen table, before heading outside to the left-hand side of the house to inspect the new water heater. You will find that you have to arrange the pipes, and once you do, you will find that the pipe that is missing is the same blood covered pipe found at the P.E. Freight Depot, solving the puzzle and unlocking the achievement.