Yes, of course, as stated by the answer from @Schwern, a quantum measurement is unpredictable (unless you have specifically prepared a state for the microscopic system, and there's been no interaction with it, in which case you recover the same state). I mention in this answer some other interesting information, and then imperfections.
Also, just to be clear, I am not commenting on the interpretation of quantum theory as either waveform collapse, or the Bohm pilot theory, or any others. Any interpretation has to agree with the observations and measurements we make in quantum theory, so local/not local, realist/not realist, are different issues than those discussed below. Whatever the interpretation (i.e., hidden variables or not), it is clear that quantum theory leads to random results in measurements.
For the radioactive case the emission of the radioactive particles happens purely randomly. The statistical distribution is a random Poisson process.
It's been possible in the last 15-20 years to measure the properties of individual photons, atoms, electrons and ions, and probably more. The tunneling electron microscope (1981) uses individual electrons to image the surface of a solid, and 'sees' the individual atoms. See https://en.m.wikipedia.org/wiki/Scanning_tunneling_microscope
Individual atoms and ions have been trapped and measured. The most intuitively clear case is observing quantum systems where the state can only be two, such as the polarization of a photon (if circular, it can only be right or left, i.e. only two states), or an electron which along any one axis can only have spin up or down, or the state of a 2-state trapped ion (rest state and one fairly stable excited state). These are done in quantum computation labs where some devices are used to use a photon or an ion and measure their states. If they are not prepared in a specific state, you will measure a random polarization or state or spin.
So the question is then what is random, when you prepare them in a specific state? Consider the electron with a spin up in the z axis. If you now try to measure the spin in the x or y axis, for that electron you carefully prepared, it will come up randomly up and down. Just like position and momentum can't be defined or measured simultaneously exactly, spin in the three directions also cannot be simultaneously defined or measured. If one is measured, the other will be totally random. And there is no way you (or nature) can prepare a microscopic system so you can measure exactly all the values.
You can also prepare a two state system in a superposition of the two states, say a superposition of up and down. When you measure it'll be 50% up and 50% down, completely random.
All of that is in principle, simply from quantum mechanics, and no matter what measurement apparatus you use. See the wiki article on quantum randomness at https://en.m.wikipedia.org/wiki/Quantum_indeterminacy.
Please note that many other systems we would think are random, can often be analyzed with a lot of data and determined as not purely random. That is true for chaotic systems, which are purely deterministic but are exponentially sensitive to initial conditions. Also often true for classical randomness, like flipping coins. Very minor and undetectable asymmetries in the manufacturing of the coin can after many runs be found to favor just a tiny bit heads over tails or the reverse, if you are patient enough, i.e., practical imperfections can often eliminate perfect randomness. But all of these are practical and classical factors. Random number generators ideally generate perfectly random sequences, but over a long time possibly the imperfections of the hardware can allow you to see some non-randomness. In a more complicated way you may prepare a quantum state, but there also can be imperfections. Thing is we've been able to create those machines and conditions in the lab well enough that we can now generate enough randomness with various processes where it could take a large number of universe ages to catch the randomness. So, in effect, and for practical purposes, it is random enough. Quantum theory seems to go beyond that, but of course in the end we build the systems, and our imprint is on them. You just sometimes would have to watch it for more time than will ever exist to see the non-randomness.