I own a 2048 x 1536 resolution monitor that has as its only input option a single VGA cable. It works very well with a standard ATI x800 video card. In theory, the resolution/amount of information that can be sent through any cable is mostly limited by the processing on either side of the cable. For instance, standard coax cable, used for antennas and SD television for years and installed in countless homes, can now be used to send multiple HD streams as well as internet signals by employing clever coding techniques on either side of the signal. This is also true of the power cables running through your house; using specially designed equipment, you can send internet and HD signals from one power outlet in your house to another using the existing cable. These are both examples of relatively low bandwidth 2 conductor cables carrying many times the amount of data they were originally intended to. A VGA cable has 15 conductors running through it, as well as a shield. A Cat5 cable that is often employed to send up to 4 streams of HD, total equivalent resolution of a whopping 7680 x 4320, has only 8 conductors! There are many other examples of this, particularly in the telecom industry, where it is necessary to find ways of using older existing cables to solve modern problems -if you're interested, just ask the humble fiber-optic cable, which recently passed a new landmark in transfer speeds exceeding its old one by 100 times!
I assure you that if you are having problems sending a 1920 x 1080 signal through a VGA cable, it is almost certainly an artificial limitation created either on the end of your PC, or your monitor(in this case it sounds like you monitor) - most likely due to the industries fascination with forcing people to migrate from one technology to another in the hopes of being the lucky one who gets to sell you something new! All of the standards in the technology we use are agreed upon at public international conventions for consortiums of electronics manufacturers. From year to year they dream up new versions of cables and encoding, and impose them on the industry in the following years. USB 2.0 or 3.0, HDMI 1.3 or 1.4, DVI-I or DVI-D, 720p or 1080p, MiniSD or MicroSD, they are all decided upon years before they are released, and roadmaps are drawn up to plan out their deployment. It's all fairly open and public actually - but collusion nonetheless in most cases to create a trickle of technology that forces people to constantly upgrade their equipment - in your case, by artificially restricting the VGA input on your monitor. It's called "planned obsolescence", and sadly you appear to be another victim! Don't worry - you're in good company(everyone)!
Interestingly, the telecom industry consortiums tend not to mandate too many changes requiring equipment and certainly cabling/infrastructure changes because the incurred costs would be paid by themselves! Thus, you end up with some of the examples above where the necessity to push the limits of existing cables resulted in some excellent innovation.
This may not have helped solve your problem, but I hope it helped explain it!