Your wife's right. And not right too.
Inkjets work by squirting ink through very very small holes in a print head. Canon and HP and several others do that by heating the ink in a chamber just behind the hole, which 'boils' a small bit of it creating a bubble, increasing the pressure and forcing out a microdrop of ink. Epson has done it a different way, with piezoelectric pulsing of the chamber behind the tiny hole.
possible failure modes
The holes are relatively easily clogged, so the fineness and uniformity of the particles in it (dyes and such) are important. So is reasonable behavior when partially dried out (not being used for two weeks during a vacation, say) and when heated or subjected to considerable pressure. Hence the cleaning cycles built into many printers (see your owner's manual for some vague and normally not very intelligible instructions).
Inks made by a knock off artist in some backroom counterfeit lab somewhere aren't likely to have uniformly ground particles, nor probably the required witches' brew of chemicals to behave well under stress. On the other hand, some third party ink producer might indeed take sufficient care to get it right for a particular type of printer / cartridge / etc. The problem in practice to how to distinguish amongst them.
Some printers have the 'print head' built into the cartridge. You get a new set of teeny tiny holes with every cartridge. With other printers, the print head is in the printer and the cartridge carries only ink. Obviously, in the latter case, a poorly made ink can be rather more expensive.
I had a long policy of buying only print head type cartridge printers for exactly this reason. And I was willing to experiment with cartridges and ink from apparently trustworthy vendors because important parts of the printer wouldn't be at risk. I cannot suggest the at home refill policy unless you can do it wearing latex gloves over a slop sink. A couple of cubic centimeters of ink can go a very very long way; you don't want to find out, but the refill kit I used did work, and the cartridges could be refilled three times at least. The very cheapest way to feed the ink monster that I discovered. Not counting the ink spilled on just about everytihint that one time... But, with printers costing less than US$100 nowadays, the cost of an experiemnt gone bad is limited, except for the time and trouble of replacing the printer damaged by the bad ink.
Some printer vendors have attempted to force their users to do the right thing (ie, buy their expensive ink -- fo rtheir own quality printing good, of course) by including in their printer measures which try to detect improper cartridges and punish the user by locking the printer to some lower performance level or some other punishment. I heard of a printer which self-destructed under when it detected fake ink cartridges., but never managed to figure out which one it was. You many find that your printer won't print any better than some draft quality if you use a third party cartridge.
With the street price of laser printers below US$150 for low end black and white, and under US$350 for low end color (shop around, at least in the US and you'll find them), there are lots of reasons to avoid inkjets, and their expensive thirst for those .cartridges. Unless, of course you need photographic quality with special archival inks, in which case you might have to use an inkjet.