If it's a black and white printer, and especially if it's a low volume unit, then third party cartridges can be okay, but the cheapest remanufactured cartridges tend to use low-quality components (especially the print drum, if contained in the cartridge rather than in the printer). The Xerox-branded cartridges I've seen appear to be somewhat better quality but my experience with them is limited.
If it's a color printer, and especially a high-volume office printer, third-party cartridges are a source of all kinds of problems. At work, we had to fight with our corporate overlords to get back to using HP OEM cartridges in our HP 55xx printers because although the four cartridges costs a whopping $350-400 each from HP, the third-party cartridges caused so many problems that the cost savings were totally lost in downtime, service calls, and cartridges that had to be prematurely replaced. My favorite was the cartridge that was somehow running out of synch with the other three, creating a color skew that looked like a belt problem.
Abacus Model 2.5 | Quad-Row FX with 256 Cherry Red Slider Beads | Applewood Frame | Water Cooling by Brita Filtration