it is possible that a very short HDMI cable might not work, while a longer one would. Here's why:
A significant number of TVs use an HDMI receiver chip that has an equalizer circuit in it. The equalizers are great in that they act as a reverse filter and allow signals that have been attenuated by a long cable to be recovered. The downside is that that equalizer can actually cause the receiver chip to be unable to read the signal if the signal is "too clean" and not attenuation. The better chips have a smart equalizer that will adjust the equalizer setting depending on how much the signal needs it.
It's possible that your TV's HDMI chip has its equalizer at a fixed setting to assume usage of a longer cable (say 6ft or longer), and thus actually has a problem with a shorter one. May sound silly, but the solution really is to use a longer cable.