The Terminator TV show The Sarah Connor Chronicles has a pretty big time travel plot hole, and the reason behind it is stupid. Like any time travel-based franchise, the Terminator series has had some trouble keeping its rules straight. Sometimes everything is presented as a stable time loop, where everything in the past and future has already happened, thus making Kyle Reese always somehow destined to be John Connor's father. Other times, Terminator leans on the idea of the future not being set, hence Sarah's drive to prevent Skynet from existing at all.
One of the clearest, and thankfully least complicated of the Terminator franchise's time travel tenets though is the idea that only organic matter can travel through time. Or at least through the portals that exist in the future where John Connor is the leader of the human resistance. That's why, dating back to the first Terminator movie, anyone who goes through has to go through naked. As for the T-800, its ability to go through is explained away by its endoskeleton being surrounded by living flesh.
For the most part, FOX's short-lived TV show The Sarah Connor Chronicles is a quite worthy addition to the franchise. It's been ignored by subsequent movies, but the story it told over its 31 episodes was well-paced and plotted, and felt like a Terminator entry. Unfortunately, there's a huge plot hole found in the first couple of episodes that's never even mentioned, much less resolved. Near the end of the pilot, Sarah, John, and new ally Cameron (Summer Glau) blow Cromartie the Terminator to pieces right before traveling through time. Yet, Cromartie's head goes flying through the portal after them. The problem is that said head was skinless at the time, seemingly undermining the entire "organic matter" premise. However, despite seeming like a significant oversight, there is actually an extremely frustrating reason that the flaw made it to screen.
The information first conveyed by Kyle Reese in James Cameron's original Terminator movie is clear, only organic material - or something covered by it - can make it through the time portal. Cromartie's head not only makes it through despite lacking anything organic over its endoskeleton, but the cyborg manages to force a scientist to build it a new body so that it can resume the hunt for John Connor. While this is a pretty glaring plot hole, it's actually not the fault of The Sarah Connor Chronicles' writers or producers.
The originally planned - and shot but not used - sequence in the pilot saw Cromartie's head fly through the time portal, but with its skin still attached. This would've prevented the plot hole, but FOX standards and practices found the image of a decapitated head flying toward the screen to be too gruesome for prime time, and demanded it be redone without the skin. Thus, an entirely avoidable plot hole still discussed by viewers of the Terminator TV show today was born. Interestingly enough, a similar plot hole exists in T-1000 being able to travel through time. In that instance, a scene was shot but cut that showed the T-1000 going through while surrounded by a cocoon of organic material. Regardless, both apparent plotholes highlight one of the many complexities inherent in the Terminator franchise.