The wheel is not wearing out. Something is wound around the axle.
The reflex when a case starts dragging is to assume the bearing has gone, because that is what a bearing failure sounds like on a bicycle. On a suitcase it is almost always something else. Airport floors, hotel carpet, the boot of a car, the pavement outside a hairdresser — every one of them is covered in long fibres, and a spinning wheel is a very effective winder. A single hair caught between the hub and the axle gets wound up turn after turn, and the next hair felts into it, and within a few trips you have a hard collar of compressed fibre packed into the gap where the bearing used to have room to move.
There is a clean way to tell that apart from a real bearing failure, and it takes one second. Spin the wheel backwards. A wrap of fibre only winds in one direction, so it works as a one-way ratchet: turn the wheel the way the case rolls and the fibres tighten and lock it, turn it the other way and they unwind and it free-wheels. A worn bearing does not care about direction — it is stiff or noisy both ways, or it has a click at regular intervals. So free one way and stuck the other is a wrap, and it costs nothing to remove.
Cut it out. Do not pull it out.
Once the fibre has felted it is not a loose thread any more, it is a compressed mat, and pulling just tears the loose end off and leaves the mat in place. Get a pair of tweezers and a small pair of scissors or a craft knife: grip a bundle with the tweezers and cut along the axle line rather than hacking at the wheel face. Work around the full circumference, in both gaps — between hub and tyre, and between hub and fork. Do not lever the wheel sideways to get at it, because the hub is thin plastic and it splits. When you think you are done, spin it: a wheel that still drags has more in there, and a wheel that spins freely but grinds has something else wrong. Then do the other three while the tools are out, because the same floor has been wearing all four.
And do not spray it with anything. Lubricant turns the remaining fibres into a sticky mat that then collects grit, and grit in a plastic hub is a grinding paste. A wheel that was dragging because of hair becomes a wheel that is dragging and abrasive. If you want to leave something behind, a trace of silicone grease on the axle after the gap is genuinely clean is fine; oil before the gap is clean is the mistake.
When it really is the bearing
Two things point at the wheel itself rather than at the wrap. It is locked or stiff in both directions, which no wrap can do. And it clicks at regular intervals as it turns — a click every few degrees is a ball race with a broken ball or a brinelled track, and no amount of cleaning will bring it back. A third sign is a wheel that has worn a flat on one side, which happens when it has been dragged locked for a while: the flat is a result, not a cause, so replace the wheel rather than trying to true it.
Replacing one is easier than people expect. Most cases use a standard wheel held by a single screw — often a cross-head, sometimes a hex socket — accessed from the outside or from inside the lining. Identify the wheel before ordering: measure the diameter and the width, and look for a moulded number on the hub or the fork, because "suitcase wheel" covers a dozen sizes. Then the detail that catches people out: check all four sit at the same height when you are done. A new wheel of slightly different diameter makes the case lean and drag on the diagonal, which feels exactly like a second failed wheel.
The handle: two failure points and one test
A wobbly handle is not one fault. Pull it to full extension and rock the top side to side while you watch the case. If the handle moves and the case does not, the play is inside the tubes: the spring pin that locks each section has worn its hole oval, or the tube wall has worn thin where the pin bears. If the handle and the whole case rock together, the play is in the mount — the plastic block screwed to the shell inside the lining — and the screws have loosened, or the block has cracked. Those are different repairs and one of them is free.
For the mount, open the lining zip, find the block, and tighten its screws; a drop of thread-locker keeps them from coming back. For the pin, some brands sell the pin and spring as a service part and it can be driven out and replaced, but on most cases the whole telescopic assembly is the unit, and it is replaced by undoing the mount screws and sliding it out. Measure the mounting hole spacing before ordering, because two assemblies that look identical can differ by five millimetres.