Ain't any such thing as "very old photos made with a computer printer" ... 'less you're conceptualizing in terms on the order of fashion trends or perishable foodstuff
Still, the archival properties of 'puter-printed photos typically is not all that great, due primarilly to the way most folks print, display, and store digital images. As with most else, you get what you pay for, and the more attention you pay to details, the better your results are likely to be. With top-shelf methodology and supplies, very good image life can be obtained - for a price and with some effort.
Generally, the accepted wisdom of the moment is that the best results will be obtained using premium OEM inks and papers specifically from your printer's manufacturer - costly, perhaps, but independent testing results bear out the conclusion. Also factors are how the image is presented and/or stored; laminating will help stave off age-fading some, protection from direct light will help some, and dark, dry, cool, air-tight storage will help even more. Finally, a digital copy on removeable media is the current ultimate in archival storage of digital images; 1s and 0s don't fade.
From a good article appearing last year in PC World Magazine:
PC World wrote:Lack of Standards Sparks Inkjet Photo Fade Debate
How long inkjet-printed photos last depend on who you ask, experts say.
Tom Spring, PC World
Friday, July 08. 2005
How long can you expect your inkjet-printed photos to last? More and more photo inkjet papers are being touted as "fade resistant" and "archival safe", but experts say these marketing pitches don't always provide good information ...
... "Ulitimately, the best way to extend the life of your images is to keep them in a photo album or even a shoe box. Displayed onwalls, images are affexted by light and air pollutants ... its best to keep digital copies of pictures on a CD or DVD"
Source (Note: 5 page .pdf download)
More than you possibly might want to know about the production, presentation and preservation of digital images:
Wilhelm Imaging Research
Lyra Research
Rochester Intstitute of Technology Image Permanence Institute