The other answers both provide useful information, but I don't feel they really answer the question fully.
After wasting an Infernal Machine to attempt to solo it, asking around on the forums, and going through the fights with a wizard capable of soloing them on MP6, I feel I now have enough of a handle on it to answer.
Short version: There are certain gear/build combinations available (presumably for each class, although I can't say for certain) that make soloing absolutely feasible. However, just because you can solo all 4 Inferno acts at a given MP level effectively does not mean you can solo the uber bosses.
Many farming builds rely upon burning through enemies quickly and efficiently, and may relegate healing or resource regeneration to health globes or time. These builds still work with regular bosses because those encounters were designed to allow you to learn the mechanics enough to avoid or mitigate most of the major sources of incoming damages (i.e. dodging the butcher's chain attacks, avoiding the ground attacks during the appropriate phases of Belial, Azmodan, or Diablo, etc.).
This is much less feasible for the ubers, due to the combined attacks.
Therefore, in order to be able to solo, you need to have some form of sustained self-healing available (either high levels of life steal or stacked Life on Hit with a properly synergistic skill). Interrupting crowd controls will help a lot, but aren't as mandatory. Uber Siegebreaker, in particular, makes some form of healing on hit of utmost importance, even when grouping.
Without this, it isn't feasible to solo the encounters.
Even with the appropriate levels of survivability, you need to have sufficient DPS, as not only are enrage timers an issue, but you have to worry about keeping the NV stacks up. If you take too long, particularly if it takes you multiple attempts, you will have to periodically go out and hunt elites to refresh your stacks, lest you risk losing your stacks partway through an uber fight (and thereby guaranteeing that they will not drop the organs you seek).