Do water hybrid cars really exist?
The idea of splitting water using electrolysis and pumping it into an engine's air intake has been around for a long time. The idea is that it's supposed to increase the efficiency of burning gasoline. However, even assuming that this is true (there are many reasons why it may not be—modern gas engines are in the range of 95–98% efficient at combustion), electrolysis is very inefficient, so I'm quite dubious of any potential gains.

Hunting around on the internet, it's obvious that a lot of people seem to think this works (and that people who say it doesn't are oil company shills or some such), but I wasn't able to find any compelling evidence to support it. How this kind of system interacts with a modern computer controlled engine may perhaps even make efficiency worse. There doesn't seem to be much academic research into it, though a Google Scholar search brings up a lot of kooky patents. This kind of thing is pretty difficult to test well—tests need to be performed under very well controlled conditions to get accurate results and is thus always subject to confirmation bias and the like.

I did find a single report done by Environment Canada on one of these products for diesel engines and they found that adding the hydrogen generator device didn't make any statistically significant change to combustion efficiency, exhaust emission rates, or fuel consumption. They calculated that the device was about 60% efficient at making hydrogen, at best, and used about 8 mL of diesel fuel every 100 km, so it doesn't really make efficiency worse to any appreciable degree, but doesn't help either. (Source: ERMD Report # 2004-032 If you search, it's on scribd, but I couldn't find any permanent sources)

In short, a water-fuelled car on its own or as a hybrid with gasoline or diesel don't seem to be possible. Water is a very stable compound and it takes a great deal of energy to break it into hydrogen and oxygen gas. At most, you can get the energy you put in back, and realistically, a lot will be lost along the way. This makes a solely water fuelled car a perpetual motion machine of the first kind as it violates the law of conservation of energy—one cannot burn hydrogen and oxygen to produce energy to make the same amount of hydrogen and oxygen consumed and still have energy left over to move a vehicle. A hybrid vehicle faces a similar problem. Because the part of the system that involves water consumes energy rather than contributes it to motion, the gas or diesel part has to produce more energy to make up for the difference. Given that modern gas and diesel engines burn almost all the fuel put into them, there doesn't seem to be any mechanism by which adding a small quantity of hydrogen and oxygen could produce significantly more energy.