If a thousand people whisper inaudibly, will the resulting sound be audible?

Yes, always.

I would like to disagree with stafusa's answer here, expanding on Rod's comment. Interference will not occur, since for whispering the sources of sound will be statistically independent.

For demonstration, let us look at two people. Person 1 produces a whisper that can be characterized by a propagating sound field $E_1(\vec{r},t)$, where $\vec{r}$ is the position in space and $t$ is time. Similarly person 2 produces a whisper $E_2(\vec{r},t)$. The overall field at a point in space is then simply

$$E_\mathrm{tot}(\vec{r},t) = E_1(\vec{r},t) + E_2(\vec{r},t)$$

since sound waves are approximately linear (at least for wave amplitudes achievable by voices).

What you perceive as 'volume' (I will call it $I$ for intensity) is the time average of the magnitude of the total signal

$$I = \langle E^*_\mathrm{tot}(\vec{r},t)E_\mathrm{tot}(\vec{r},t)\rangle.$$

That is, your ear is averaging over very short fluctuations in the signal. We can then expand this in terms of the two people's signals to get

$$I = \langle E^*_{1}(\vec{r},t)E_{1}(\vec{r},t)\rangle + \langle E^*_{2}(\vec{r},t)E_{2}(\vec{r},t)\rangle + 2\langle E^*_{1}(\vec{r},t)E_{2}(\vec{r},t)\rangle.$$

So far, this is completely general. Now we assume statistical independence of the sources, which makes the last term zero:

$$I = \langle E^*_{1}(\vec{r},t)E_{1}(\vec{r},t)\rangle + \langle E^*_{2}(\vec{r},t)E_{2}(\vec{r},t)\rangle.$$

So the overall intensity is simply the addition of the two whisper intensities.

The amplitude of the sum of $1000$ equally loud uncorrelated noises will be about $\surd1000$, or approximately $32$, times the amplitude of a single noise. That might be enough to make an inaudible whisper just audible. Consider the practicalities, though. The people cannot all occupy the same spot. If dispersed, most of them will be too far away to hear. Even if crowded together, their bodies and clothing will make an excellent sound-absorbing medium. In probability, all you will hear is the involuntary sound occasionally emitted by a single individual.

Yes, and if done carefully, not only audible, but also understandable.

[Update: indeed, see the answer from Floris - include audio files to prove it!]

For example, they should whisper exactly together only if they are equidistant from the hearing point - for an arbitrarily chosen point, they should whisper with small delays among them, so that the sound reaches the point in sync, interfering constructively.

Edit: That's so if, besides being audible, it's desired the sound is also understandable. As many pointed out, even whispered random noise will lead to increased volume.

Also, "done carefully" can be achieved in ways other than above, which is just an example. Another way is whispering/talking slowly: like when students greet in unison an incoming teacher, or people in a auditorium respond to an entertainer request.

And, lastly, probably it really has to be only "somewhat carefully", since the increase in volume also happens (and the words can often be understood) when the public in concert, a choir, or a group of churchgoers sing together.

An example of a choir whispering might convince the naysayers ;-) https://youtu.be/yaNeIgBZSUE?t=89