Does the quality of ink matter for sublimation?

All inks are not created equally. Each ink manufacturer has their own recipe for creating ink. Different inks produce different levels of quality and different ranges of color. Some inks may require additional saturation to produce the proper color, whereas others can achieve great results with minimal application.

But there is more to it than just the ink. Print quality is affected by the combination of the printer, the paper, and the color management software. Some companies simply buy ink, label it, and sell it for off-the-shelf printers. Other companies actually manufacture their own ink (in-house) in conjunction with their own printers to ensure the best possible color and quality of images. 

Think of ink like paint. You can buy average-quality paint that gives you decent results but might take multiple coats to get the job done. And it may only last a few years before fading or peeling. On the other hand, there are many high-quality paints that can do the same or better job with just one coat and last a lot longer before any degradation of quality.

   —Sawgrass