In my own experience, transitioning from "old school" dual blade razors and/or electrics, using the Gillette Mach 3 was a revelation in how much closer and smoother a shave I could get with little effort, and how fewer times I would tend to cut myself when shaving in a hurry.
I also had great luck using these same blades to shave my head bald.
Which is where things start to diverge a bit…
In more recent years, I've received test samples of the Quattro and whatever the 5-bladed monstrosity is called. For facial shave, the smoothness/ease of use did seem to increase incrementally by number of blades.
The cutting effectiveness, however, was reduced by one simple fact – the blade heads are only slightly bigger than their 3-bladed counterparts, which means thinner and/or more angled blades in each cartridge. For me, this meant the blades would dull much faster than with the Mach-3. Just to give you an idea, I usually pull about 1 week out of a blade (pretty coarse hair) doing just my face, or a single use between scalp and face (then it gets dull/nicked enough that subsequent uses turn me into a bloody mess).
Because of the sheer number of blades, the newer units also seem somewhat less "flexible", which also made using them to shave the top of my head basically impossible.
Overall, I find the 3-blade units are the best overall price-performance value for any type of shaving. If I was going to keep a separate set of blades for my scalp, I'd probably scale back to the "Sensor Excel", a dual blade unit, but use the "HeadBlade" to hold the cartridge (versus the standard "stick" style razor handle).
And if I had a lot of money kicking around, I'd probably then 'upgrade' my face razor to the Quattro or above (but at that stage, it might make more sense to just get a pro barber shave given the huge cost per cartridge).
To sum up: 2-3 blades allows you to cut more effectively without pressure/angling of the blades that may otherwise cause cuts… And 4-5 blades will certainly increase the feeling of "closeness", at least right after the shave.
Is there any real advantage to using a razor with more blades, or is it just marketing mumbo-jumbo?