And they usually aren't.
I'm a radiologist and computers have definitely improved patient care and saved the hospital money (or alternatively made the hospital more money) in our field. From digitized images and the ability to outsource to overnight coverage to voice recognition to get turnaround for finalized reports in an hour it has undoubtedly worked. And that's with in most cases only fair implementation of a computer system.
With most hospitals, the problem is that they like to do a piecemeal transition. Digitize a subset of notes and vital signs, half the time what you need isn't there so you have to look through the paper chart AND the computer chart. Or the vital signs are only half in the computer and half on a chart, so nurses double their workload. And when it's set up, they do it with an IT-centric interface that doesn't make intuitive sense to most users. When I use them I can see through my background in computer science and engineering why things are done a certain way, but it doesn't make any sense to physicians, nurses, etc.
Then they add in a new piece, such as more vital signs (but in a different section), some dictated notes, some linking to the outside. Outpatient notes are digitized, inpatient notes are still handwritten, etc. ED notes are separate, with their own system. It's a complete mess. This method is a waste of money and time, all for the sake of early deployment of a suboptimal system and minimal re-training of the staff to use a new system.
The VA had a decent attempt with CPRS. They digitized everything - from physician admission notes to clergy notes. At least everything is in one place, but people are overwhelmed with data and it's too easy to copy and paste incorrect or inaccurate information. The interface is also suboptimal (graphing lab values involves selecting a range of tests, building a worksheet, etc. much like you'd expect an engineer to make it for maximum flexibility, but minimal ease of use). And connecting to other VA systems is hit or miss.
Perhaps the best method is to build a new hospital from the ground up. All patient records get digitized (scanned, at least, if not run through some OCR). Have a tightly integrated medical record system developed in collaboration with health care practitioners. That would save the hospital money, in the long run, compared to them starting from scratch with paper records.