Actually it's not that a simple matter as simple output voltage clipping. Most audio amplifiers are AC coupled and only the AC parts of a signal actually carry power. So it would be perfectly possible to have a sine signal with its samples all between 32000 and 32767 that would do no damage at all to the speakers, because the AC power it carries is low (-38.63dB(FS) to be specific in your case).
But it's not just the total AC power that matters, it's also the power density that matters. So for example a sine has a very high power density (in fact if you do the math, the power density for a pure sine is infinite). This is turn means, that whatever resonances the system has, that overlap with the frequency of the sine, they will get excited and can resonate toward mechanical breakdown (damaged speaker). A low power density signal, like noise, where the total output power is spread over the whole spectrum (or the spectral response of the system in question) and below the maximum rated power of the speaker is much less likely to do damage.
Now we have to look at clipping. Analog clipping is not just some random noise, it's in fact a harmonic distortion. And as such it can redistribute the power in the spectrum. And digital clipping actually causes an integer overflow, which in fact adds further power to the signal.
But it goes further: The output of a DAC is a voltage. A voltage by itself does not carry power, you need an amplifier to do that. So the first step after the DAC and after the reconstruction filter (which also acts as a DC block) is a transimpedance amplifier that turns the voltage into a power signal; what this means is, that for a signal of a certain power that's going into a certain load it gives rise to a specific RMS voltage on the load terminals. In audio the reference power level is usually 0dB(m), i.e. 1mW – on the typical load impedance of 1kOhm this corresponds to a RMS voltage of 1V at the load (and higher voltage at higher impedance, lower voltage on lower impedance loads).
In a perfect amplifier the impedance is independent from the input signal and its output load. In a real amplifier though the impedance can vary depending on the signal and the output load. And the impedance usually also depends on the frequency of the signal.The effect is, that an amplifier introduces further harmonic distortions, if the signal puts too much power onto its input. If this happens in the preamplifier stage this can drive an output stage outside the specifications of the connected loads.
You see, because distortion happens mostly if you drive an amplifier close to it's maximum capabilities the usual approach is to design the amplifier some comfortable margin to the needs for the connected load; typically you use a margin around 6dB, so for a 6W load you'd choose a 12W amplifier, so that no matter what input signal the amp gets, the total harmonic distortion stays within safe limits. To protect the load you then choose the amplification factor so, that for a reference power input signal the output power is lower than or equal to the power for which the load is rated.
Now say, you're a cheapskate engineer, maybe with little analogue electronic design experience. So you see your speakers have a maximum rating of 6W. So you do a parametric search and come up with amplifier ICs from the usual suspects (Maxim, TI, ST); they're rated a 5W, 10W, 15W...; obviously cannot drive those 6W speakers at maximum power (it will badly distort), but 10W should do the trick, right?
Well, now, because you forgot the total harmonic distortion: Which means that if you raise the input signal power by 6dB (i.e. double the power) the output of an amplifier may rise well over 6dB. So lets say you had first a signal output power of 2.5W for a -6dB(FS) signal, then you raise it to 0dB(FS) (which would be your volume slider in the control applet at 100%), theoretically the output signal should now be 5W. But because of the underrated amplifier the output distorts and the actual power going into the load rises well above their maximum 6W rating.
TL:DR; Dell did mess up their amplifier design, but not by overrating the amplifier, but by underrating it. If they had chosen a properly rated with margin (so that THD stays low) amplifier and chosen the gain so that a 0dB(m) line signal would have produced 6W output power (about 15dB gain) the speakers would have been safe.