About a third of the way through when Dr Fleinhardt joins the agents at the FBI, when the camera zooms in on Dr Fleinhardt a tall random agent walks from the back of the office toward Fleinhardt and then when the camera zooms out the same agent is seen appearing from the back walking towards Fleinhardt.
The opening set of numbers with the running stopwatch is flawed. It fades to Ron running with the stopwatch time at about 21:41 minutes with Charlie stopping the stopwatch less than a minute later. At the speed of 12.9 MPH, it would take 28:54 minutes to run 10,000 meters which is significantly longer than Ron's supposed time. At the time this episode aired, the fastest 10k time was 26:17.53 held by Ethiopian Kenenisa Bekele.
When the agents go in to question Doug Windom, in the long shot he is holding the cutting touch to far from the metal he is cutting. In the closer shot he is holding the touch much closer, but the metal isn't yet ready to cut. If he put the heat on his work the moment the scene began, by the time of the close-up he would be ready to begin the cut.
When the thief is climbing the rope, twice he slides both hands up the rope at the same time which is impossible since there would be nothing to keep him from falling when he released his grip. The rope is not attached to any part of his body to provide the support that would be necessary to accomplish this feat.
Even though there are LIGO labs at Caltech and MIT the actual LIGO detectors which are in underground tunnels are located near Hanford, WA, and east of Baton Rouge, not at Caltech, so there is no tunnel to be broken into.