What's possible:
* Manipulating iron (Fe) with magnets. Fe is ferromagnetic and, therefore, can be controlled (to some extent) using magnets.
* Magnetic micro robots. It's been done, albeit working mainly on 2D.
* Lots of simple robots working together to accomplish complex tasks. I've spent a year doing (basic) research in Swarm Robotics I can say that most of the stuff about controlling lots of robots like in Big Hero 6, discussed on Film Theory, is quite possible. The battery problem (mentioned in FT's video) can be mostly solved if the nano-machines are powered by the magnetic fields inside the controller, except for the problem of the strength of the fields. (later discussed in the problems part)
* Manipulation of magnetic fields. I'm not an expert in magnetism but magnetic fields can be redirected and custom-printed magnets already exists (though not at the scale and level you're using).
* Magnetic tweezers are a thing, apparently.
* Manipulating things at a molecular level. I'm not an expert at molecular physics but manipulation of individual atoms and molecules is very difficult. However, it can be done, as demonstrated by IBM here. You do actually need to be able to control your magnetic fields and robots at the same scale though.
Problems:
* Magnetic fields weaken following the inverse square of the distance. This means the strength of the magnetic forces decrease quite dramatically as the distance between the magnet(s) and the target robots increases. Objects 2m away from the magnet(s) will be pulled with a force 1/4 of one 1m away.
* "Given permission to let loose, the robot could eventually reach titanic scales." Using just the magnetic fields? Probably not. While you can extend the magnetic fields' affects by linking lots of robots directly together (hold a magnet, put a paperclip on the magnet, then chain more paperclips onto the attached paperclip(s) without them touching them the magnet directly and you'll see), precision control would probably require your magnets to reach most, if not all, of your robots for good control. You can stick more stuff onto your chunks of robots but maybe not actually control them. At least not very well.
* Manipulation of atoms. While you can push some molecules around, it'll take a lot of energy to make them bond, and probably even harder to split them. You can gather lots of water molecules to form steam, but will need a good amount of pressure to turn the steam into water, and even more to form ice. 284
With nanobots at the size of a few dozens of iron atoms, you probably can't split those water molecules to make Hydrogen and Oxygen.
* Controlling thousands or millions of robots with just magnetic fields. In Swarm Robotics, commands are often sent to low-intelligence robots, each of which can do some calculations. Since all you're using is a few dozen Fe atoms for each robot, we'll have to rule that out. Instead, you'll have to:
* Make the robots align with your magnetic fields. This means you'll have to make your magnetic fields the shape of an arm if you want your robots to form an arm. While magnetic field manipulation is possible, it doesn't get to that level (as far as I know). You can, of course, use "sci-fi" as a reason to be able to make such complicated (and may be even impossible) magnetic fields.
* Use your magnetic fields in pulses to form some shapes. I don't have a solid idea of how this can be done, but I'm thinking of something similar to the Fourier Transform (google that). You can (may be) make lots of magnetic fields in pulses that somehow sum up to form the shapes you want.
Suggestions:
* Acoustic Levitation. You may add this to help with controlling the robots. It does also have the same "inverse square law" problem as magnets though.
* Directly control stuff with the magnets if you can. If you can already control the robots with that kind of precision, you can also control lots of other things using your magnets without having to let the robots try to move the particles.
* Use the robots to make circuits. You can use wireless power by making generators with the robots and supplying them power with your changing magnetic fields. This way, you can wireless transfer electricity and create extra magnetic fields using the robots. (Google "how generators work" for more detail.)
* Use more complicated robots. Controlling thousands of something equivalent to door hinges with just magnetic fields is a big stretch. If you make the robots low-intelligence, however, they can be controlled much better and more realistically. With low-intelligence nanobots, you can levitate them with the magnet(s) and control them digitally similar to real robots.
Disclaimer: I only have real knowledge of Swarm Robotics. 284