Is it possible that an instance of a plant could just be a dud. As in even with great care it is never healthy?
Through natural propagation (ie. seeds) sure - most mutations will be relatively benign (e.g. different color or different weather tolerances), but some are going to result in plants with low viability. Similarly, some are going to result in plants which are much better suited to their local environment. The bad ones die, the better ones do better. A cornerstone of evolution.
Propagation by cuttings (i.e. cloning) is a different matter as the clone is going to be genetically identical to the parent plant.
In my experience, yes. Though it's hard to say if it is genetics or something just went awry when the plant got started. I've started batches of, say, broccoli where all of the plants started from the same batch of seed, grew in the same media, under the same lights, in the same temperature, with identical watering. And they all do great -- except one that just doesn't do what it's supposed to.
This is why, if you're saving vegetable seed, you need to carefully "rogue out" the off-types. Because there will be mutations or cross pollination and you don't want your seed stock to get polluted.
Similarly if you're propagating plants vegetatively. If some mutation occurs and you end up with an individual that isn't thriving (or otherwise does not have the properties you're looking for -- perhaps even just the wrong coloring), you wouldn't choose that one to propagate.