Cables have speed ratings, and there are only two speeds of HDMI cables: Standard and High Speed.
Any high speed cable is required to pass 1080p video across it regardless of the length. Standard speed cables are only required to pass 1080i video.
In my experience, even cables only rated as standard speed, often handle 1080p video just fine, but they are not guaranteed to do so.
That said: There are certainly cables which are 'standard speed' rated only at lengths of 6'. These ultra-thin cables from different manufacturers typically handle 1080p just fine, but aren't guaranteed to do so at that length.
For example:
Extron HDMI Cables & Adapters - HDMI Micro Series
The 1.5 and 3' micro HDMI cables are high speed rated, but the 6' cable is clearly labeled as standard speed.
Getting a larger gauge of cable, to replace an existing high-speed cable will not give you better zeroes or ones, so replacing the cable is pointless and won't help things. If you are having video dropouts, or other actual issues, then the cable may be poorly constructed, have a defect in it, be picking up some interference, etc. There are dozens of reasons why a HDMI cable may not work properly, but on shorter lengths, I have almost never run into issues.
In fact, on shorter lengths, I put together a review of the Parts Express Super Slim HDMI cables which I found to be excellent and a class value leader in price to performance.
super slim hdmi cable - Parts Express Ships Fast and Ships Free.
I use the cables with HDBT extenders, and other devices to my A/V receiver, and directly to/from components as allowed within my system, and have had zero, and I mean ZERO issues when using them. I probably have 10 or more within my system at the moment. Plans are to include a likely upgrade to put in another dozen or more.
I rarely use, and rarely recommend thick HDMI cables except for long haul runs of 25' or more. Then I typically recommend 22AWG to reduce the number of problems.