does my cable box support an HDMI connection?

Hard to tell. I think some DCT6416 had HDMI and some had DVI, depending on exactly when they were released. Thats because HDMI was released right around the time those boxes came out.

Are you asking if your box has a HDMI port? Or are you using it and it's not working? If so, might have to check user settings. See page 8-9 of the manual.

Also, if still having problems, you can check the diagnostics. With the cable box on and connected normally, press power,select,select on the front of the box. You should see a screen like below. Then use the cursor up/down to bring the > to D11 and press select. Should give you the list of the different ports. There will be two columns. One for enabled, and one for used, or something like that. As long as your Yes/No and not No/No, you should be all set. 

Yes, the Motorola DCT6416 III does have an HDMI Output

http://www.motorolasolutions.com/web/Business/_Documents/static%20files/DCT6400%20Data%20Sheet.pdf?pLibItem=1&localeId=33

http://www.motorolasolutions.com/web/Business/Products/TV%20Video%20Distribution/Set-tops/QAM%20Set-tops/Digital-Analog%20QAM%20Set-tops/DCT6400/_Documents/Static%20Files/DCT6400%20Series%20User%20Guide.pdf?pLibItem=1&localeId=33

Yes, my box has an HDMI port but when I try to connect to TV all I get is no or weak signal message.    Tried 4 different ports on TV...

coxcabletvques1 said:

Yes, my box has an HDMI port but when I try to connect to TV all I get is no or weak signal message.    Tried 4 different ports on TV...

yes, 4 hdmi inputs

coxcabletvques1 said:

yes, 4 hdmi inputs

Nice! Model number?

Older Samsung

UN46B6000

ColleenD said:
coxcabletvques1, 

If you've verified that the HDMI port is not disabled, tried at least 2 HDMI cables and all the ports on your HDtv to no avail, it's possible the HDMI port in the DVR has stopped functioning. It would be necessary to swap out the DVR at your local Solutions Store or have a technician scheduled to visit your home to swap it. Hours of operation and the location of your closest store can be found here http://ww2.cox.com/aboutus/contact-us/cox-centers.cox by ensuring your location is correct in the top right corner of the page.  

Indeed. The HDMI circuitry on Motorola and Cisco DVRs and receivers is...uh...less than robust. Why is that?

ChrisL said:
That's hard to say, it could be the result of 1st gen technology however I'm not aware of any specific issues that were identified.  I did actually run into a similar issue with an older box and my 1st HDTV and the fix was as simple as a box swap.  The component cables are also an alternative as how they're capable of supporting what is being broadcast to the fullest.

True, but component cables are analog, causing two things:

1) an extra D->A and an extra A->D conversion, and 2) the possibility of introducing analog video noise to the signal. Keeping the signal digital all the way to the TV eliminates both of those problems. Plus, an extra set of analog AUDIO cables must be used with the component VIDEO cables.

Depending on your TV, I have actually seen some that look better with component then HDMI. Alot of it has to do with the cheap circuitry in the box and some TVs. Your right of course about HDMI being the better connector overall though. I think either way, the OP would be better off swapping it for a newer box if possible. That would fix the issue if it was a hardware issue, a firmware issue, or a setting issue (since most knew boxes default to HDMI).