Physicist: With very few exceptions, yes. What we normally call “random” is not truly random, but only appears so. The randomness is a reflection of our ignorance about the thing being observed, rather than something inherent to it.
For example: If you know everything about a craps table, and everything about the dice being thrown, and everything about the air around the table, then you will be able to predict the outcome.
Not actually random.
If, on the other hand, you try to predict something like the moment that a radioactive atom will radioact, then you’ll find yourself at the corner of Poo Creek and No. Einstein and many others believed that the randomness of things like radioactive decay, photons going through polarizers, and other bizarre quantum effects could be explained and predicted if only we knew the “hidden variables” involved. Not surprisingly, this became known as “hidden variable theory”, and it turns out to be wrong.
If outcomes can be determined (by hidden variables or whatever), then any experiment will have a result. More importantly, any experiment will have a result whether or not you choose to do that experiment, because the result is written into the hidden variables before the experiment is even done. Like the dice, if you know all the variables in advance, then you don’t need to do the experiment (roll the dice, turn on the accelerator, etc.). The idea that every experiment has an outcome, regardless of whether or not you choose to do that experiment is called “the reality assumption”, and it should make a lot of sense. If you flip a coin, but don’t look at it, then it’ll land either heads or tails (this is an unobserved result) and it doesn’t make any difference if you look at it or not. In this case the hidden variable is “heads” or “tails”, and it’s only hidden because you haven’t looked at it.
It took a while, but hidden variable theory was eventually disproved by John Bell, who showed that there are lots of experiments that cannot have unmeasured results. Thus the results cannot be determined ahead of time, so there are no hidden variables, and the results are truly random. That is, if it is physically and mathematically impossible to predict the results, then the results are truly, fundamentally random.
What follows is answer gravy: a description of one of the experiments that demonstrates Bell’s inequality and shows that the reality assumption is false. If you’re already satisfied that true randomness exists, then there’s no reason to read on. Here’s the experiment:
The set up: A photon is fired at a down-converter, which converts it into two entangled photons. These photons then go through polarizers that are set at two different angles. Finally, photo-detectors measure whether a photon passes through their polarizer or not.
1) Generate a pair of entangled photons (you can do this with a down converter, which splits one photon into an entangled pair of photons).
2) Fire them at two polarizers.
3) Randomly change the angle of the polarizers after the photons are emitted. This prevents information about one measurement to affect the other, since that would require that the information travels faster than light.
4) Measure both photons (do they go through the polarizers (1) or not (0)?) and record the results.
The amazing thing about entangled photons is that they always give the same result when you measure them at the same angle. Entangled particles are in fact in a single state shared between the two particles. So by making a measurement with the polarizers at different angles we can measure what one photon would do at two different angles.
It has been experimentally verified that if the polarizers are set at angles 
We can do two experiments at 0°, 22.5°, 45°, 67.5°, and 90°. The reality assumption says that the results of all of these experiments exist, but unfortunately we can only do two at a time. So C(0°, 22.5°) = C(22.5°, 45°) = C(45°, 67.5°) = C(67.5°, 90°) = cos2(22.5°) = 0.85. Now based only on this, and the reality assumption, we know that if we were to do all of these experiments (instead of only two) then:
C(0°, 22.5°) = 0.85
C(0°, 45°) ≥ C(0°, 22.5°) + C(22.5°, 45°) -1 = 0.70
C(0°, 67.5°) ≥ C(0°, 45°) + C(45°, 67.5°) -1 = 0.55
C(0°, 90°) ≥ C(0°, 67.5°) + C(67.5°, 90°) – 1 = 0.40
That is, if we could hypothetically do all of the experiments at the same time we would find that the measurement at 0° and the measurement at 90° are the same at least 40% of the time. However, we find that C(0°, 90°) = cos2(90°) = 0 (they never give the same result).
Therefore, the result of an experiment only exists if the experiment is actually done.
Therefore, you can’t predict the result of the experiment before it’s done.
Therefore, true randomness exists.
As an aside, it turns out that the absolute randomness comes from the fact that every result of every interaction is expressed in parallel universes (you can’t predict two or more mutually exclusive, yet simultaneous results). “Parallel universes” are not nearly as exciting as they sound. Things are defined to be in different universes if they can’t coexist or interact. For example: in the double slit experiment a single photon goes through two slits. These two versions of the same photon exist in different universes from their own points of view (since they are mutually exclusive), but they are in the same universe from our perspective (since we can’t tell which slit they went through, and probably don’t care). Don’t worry about it too much all at once. You gotta pace your swearing.
As another aside, Bell’s Inequality only proves that the reality assumption and locality (nothing can travel faster then light) can’t both be true. However, locality (and relativity) work perfectly, and there are almost no physicists who are willing to give it up.
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