Coil whine usually doesn't affect either the PSU's or the components' operation, at least not to a measurable degree, since the resonant frequencies are typically such that they're easily damped in other components, such as capacitors and other filter coils. I believe all manufacturers are actually aware of this issue, and are doing what they can to deal with it, most of the time.
Namely, there are three ways coil whine will develop. There's "self-whine" to which every coil under the Sun is susceptible (including various types of transformers) by it's very nature, and there's resonant/induced whine, which is a byproduct of resonance between VRMs on the motherboard and/or the graphics card, and the PSU's coil(s) and/or transformer(s). Let me elaborate further:
As the current passes through a coil, it creates a magnetic field, which in turn induces a current in the coil such that it tends to cancel out the change in the initial current. So if there's a constant 1A through a coil, then it jumps to 1.1A, the change in magnetic field will induce a current of -0.1A (meaning 0.1A in the opposite direction), restoring the net flow to 1A. This is how coils remove unwanted ripple/noise from the DC output of a PSU, or a DC input into a VRM.
Both the length of the coiling wire and the coil loop diameter are parts of the inductance equation, and are a variable just like the inductance is, and not constants. Well, theoretically they are constants for a given coil when it's effective inductance is calculated, but in real world, where approximations amount to a wrong result, the coil will shrink and expand under the influence of magnetostriction.
Self-whine or coil noise can be twofold - physical and electrical.
Physically, high frequency switching used in PSUs (50-150 kHz) will make the coil vibrate (from all the rapidly succeeding shrinking and expanding) at a lesser frequency, typically from one quarter to one eight of the switching frequency. This is sometimes well inside the audible range (~20 Hz - ~20 kHz, typically 30 - 18k). The lower frequency vibrations are a consequence of the finite velocity of current (rather, electrons) and the finite speed of expansion/shrinkage propagation through the coil. Not only that, but both the wire and the core are shrinking/expanding, and at a different rate and amplitude, so until everything aligns properly (rate and speed of shape change with the rate of propagation of the deformations), there will be no audible vibrations. This is part of the story.
Electrically, as the coil loops are moving and the core changes shape, both travel inside a varying magnetic field, which causes additional self-induced currents to appear. These are mostly damped out by other filtering elements, due to their very low magnitude and their relatively high frequency, but sometimes they manage to get to an amplifier in a sound card, for example, and show up as audible noise in the sound (sub)system. Additionally, every coil is a (poor) antenna for high-frequency signals (voltage changes), and it radiates those signals out into wires and PCB traces. There they are induced back from electromagnetic emissions into current and possibly amplified as per above.
The kicker is that physical noise can (and does, in larger inductors) cause electrical noise, and vice-versa. Further, any wire or other form of conductor (like a PCB trace) is also an inductor, albeit a poor one.
Resonant whine can develop between any two oscillatory systems, which coils are all by themselves, as is practically any circuit that contains them. VRM circuits on motherboards, graphics cards, hard drives, etc. pretty much always contain at least one. In order to have an electrical oscillator, you need an inductor and a capacitor. All inductors are also (poor) capacitors, and this doesn't present a problem at low frequencies, because they "see" capacitors as open circuits. Self-capacitance is a problem at high frequencies, exactly the situations where you'd want to use coils in the first place... When two coupled coils (either connected via wires and traces or magnetically coupled, or when the EM radiation of one permeates the other) reach very similar electrical self-noise frequency, the parasitic signals they produce may be (and usually are) amplified exponentially. This can, in rare cases, actually pollute the DC input/output, and there are actual cases in practice. There are some Sirfa-made PSUs in which simply moving output wires away from a regulator coil makes the PSU output voltage significantly less noisy. I still consider this a rarity, though, and it can be solved by putting a simple EMI shield (a piece of isolated metal sheet) around the offending coil or between it and the "polluted" area.
Coil whine can be lessened to an acceptable degree with a relatively simple fix. Just dampen the physical side of it by gluing or caulking down the coil, so that it's vibrations are absorbed. Another way is changing the current/voltage frequency, which is never easy, as it affects the electrical design of the device in question, or use a different coil. This could be a coil made of different materials, or of a different size, or even a different shape. I've seen coils made in the shape of the number 8 (or the infinity sign, if you need to be geeky about it), that produce significantly less noise than standard toroidal coils. I can't say how much this would add to the price, however. And let's not forget that transformers are, in effect, simply big-ass coils, and their whiny nature is much harder to deal with...
As for why coil whine would develop in time, instead of right from the get-go, well... Perhaps the dampening glue/caulk "breaks in"? Or maybe the coils very slightly change their basic (at-rest) shape, such that their resonance pattern shifts into the audible range? Who knows, it's a very complex phenomenon.
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