Same problem here with a Specter x20G Naga III monitor. A few days ago I tried to set up a new Mac mini for my wife. There was no video from the HDMI to DVI converter. I plugged the old Mac mini (circa 2007 but updated to Snow Leopard) in and the video worked just fine. I called Apple Care, they made an appointment with me at the Genius Bar, I took the new mini in and, using the same display adapter, it worked fine with their monitor. While installing the software updates at the Genius Bar, that mini kenel panicked. So they issued me a new one.
I brought the new mini home, plugged it in, and I had no video just like the first one. I tried a different DVI cable. No love. I tried another monitor with the HDMI to DVI converter and the video came through just fine. So I went back to the Apple store to pick up a mini display port to DVI converter. It didn't work on the Mac mini. It also did not work on my Macbook air.
My conclusion is that some DVI monitors just don't work with Lion. (Lion being the one commonality between my Macbook Air and my wife's Mac mini.) I ended up buying a new monitor. It works fine with the new Mac mini. I took the old monitor and set it up as a second display on my daughter's Mac mini which is still running Snow Leopard. It works great.
No HDMI and DVI are both digital signals there will be no degradation in quality between the two. If in fact they are compatible with each other as referenced on Wikipedia.
HDMI is backward-compatible with single-link Digital Visual Interface digital video (DVI-D or DVI-I, but not DVI-A). No signal conversion is required when an adapter or asymmetric cable is used, so there is no loss of video quality.[5]
Both are much better than VGA. VGA is an analog signal. There are noticeable differences when using a VGA connection vs a pure digital one. Everything will appear to have a slight fuzziness, it's not horrible but if you have good vision you can see the difference up close. With pure digital, the text, images, etc with all be sharper. This is a side effect of the analog to digital conversion that the LCD monitor has to do. Additionally as part of that conversion process most LCDs have an auto correction button to dynamically adjust the VGA signal to fix over-scan and other minor adjustments to the VGA source. Something that you don't ever have to do with a digital source. I would avoid buying a monitor that only has VGA input, spend $20 to $50 more and get a better display that has at least one digital input source, either DVI or HDMI.
Confirming Source for LCD Analog vs Digital inputs.
I have had bad luck with 2 Mini-DVI to VGA adapters purchased from Monoprice.com that had Red, Green, Blue hewing due to poor wiring. For example, I had to slightly twist the cable to get the picture color just right. They replaced one and then the second one did the same thing, I am due for a third try... However my mini-DVI to HDMI adapter from them works flawlessly and has since day one.
Best Answer
Not only is this not impossible, but products that do this exist. DisplayPort (and therefore mini-displayport and thunderbolt) supports DVI and HDMI output over passive adapters using Dual-mode DisplayPort. Dual-mode DisplayPort only supports single-link DVI or HDMI. Single-link DVI is limited to 1900x1200, but single-link HDMI can go up to 2560x1600@60Hz with HDMI 1.3+, and bigger on new HDMI versions. In the DisplayPort 1.1 spec, VESA introduced a spec for dual-mode DisplayPort to support HDMI 1.4 output, using "Type 2" adapters.