many variables here.
1) Your cable box needs to output HDTV via either HDMI or red/bue/green component. S-video or composite (yellow RCA jack) cables won't do it. (this is for anyone else who might be reading this)
2) All HDTV is digital, but not all digital TV is HD. Depending on your service provider, you may need to upgrade your service, and possibly rent a HD cable box, to gain access to HDTV channels. I have Comcast and I had to. I got dsome digital but no HD w/o the box and service upgrade
2.5) If you have a built in ATSC tuner, you might be able to receive some HDTV over the air.
3) If you are receiving either 720 or 1080 HD, you WILL be able to, without any doubt, discern the difference between that and 480 SD. PBS or Discovery TV offer eye-popping HD shows. But, shows made for HDTV fill the entire screen. They are not letterboxed, nor are there bars on the sides. "regular" shows shown on a HD channels will have bars, though.
I'm not sure what "dot by dot" mode is. Perhaps others might be able to offer assistance but from what you describe about "the office", you MIGHT be getting HD but I don't know enough about your TV or service to offer a definitive answer.
With my HDTV setup, most SD shows and virtually all commercials fill the screen top-to-bottom with bars up and down the sides . A few shows have some letterboxing all around, but not many.