When the filter material becomes filled and more clogged, the aerobic bacteria have less oxygenated water flow to break down ammonia and nitrite into nitrate, which plants use, but nitrate can build up in the aquarium if water changes are not performed. However, as aerobic bacteria dies off in the absence of oxygen due to the reduced flow, it is replaced by anaerobic bacteria, which further breaks down nitrate and other wastes into free nitrogen (which mostly leaves the aquarium as a gas) and into other nutrients that plants can use, so anaerobic bacteria isn't all bad. But another product of anaerobic bacteria is hydrogen sulfide (you know, that "rotten egg" or sewage smell?) and other toxins, which was the cause of "sudden tank-death syndrome" way back in the early days of aquarium keeping, before UG filters, power filters, and before affordable artificial lighting was sufficient for growing many plants. (Basically what was happening was a substrate--usually fine gravel or sand--was too deep and went undisturbed for a long period, and then something dug a little too deep and hit a pocket of anaerobic bacteria, releasing the hydrogen sulfide and killing everything in the aquarium.)
That said, in his book, "Encyclopedia of Aquarium Plants", Peter Hiscock recommends allowing part of the filter medium to become partly anaerobic by alternating the rinsing of half the medium and leaving the other half " dirty" and somewhat clogged, and rinsing that half the next time the filter is cleaned, leaving the other half unrinsed. This is said to naturally provide nutrients for the aquarium plants. Whether you prescribe to this train of thought or not probably depends on your personal degree of anal-retentiveness, but I imagine a small degree of anaerobic activity in a filter that is regularly cleaned may be beneficial to plants, but wouldn't do a thing for animals, other than potentially harm them. Besides that, most hobbyists here are artificially providing nutrients by dosing in controlled measure, although if you wanted to achieve a balanced, fully closed "natural" aquarium system without any additions other than light and topping off the water level, it might have merit.
As has been stated above, it all depends upon many variables, including your intentions for the aquarium, the animal stocking load, the plant stocking, how much you feed the animals, what scavengers you have, etc.
I just bought a setup that included two Fluval 305s, which I immediately put on a 55 without cleaning. (The 55 is newly set up, specifically for three plecos that came with the new setup, so I wasn't concerned about introducing anything harmful to an established aquarium, plus I wanted the ready-made nitrogen cycle for the plecos.) When I noticed the flow was practically nil, I broke one down to see what the problem was. The filter pads were so clogged and full of what can only be described as "mud" that a filter basket filled with filter pads in the bottom and ceramic bio-medium on top of that actually floated in the bucket for several minutes, until I got impatient and pushed it under. Now THAT'S a dirty filter!