Absolutely!
In fact, in my opinion, the most important "math skill" that should be taught in conjunction with, and using, word problems is checking whether the answers make sense. This is an absolutely invaluable part of making any practical use of mathematics, as opposed to just blindly applying formulas for the sake of passing an exam.
There are several parts to this skill, which all should be explicitly taught and reinforced at every level, such as:
Dimensional analysis: A bicycle does not move at a speed of 15 kg/m². Any such answer is simply nonsense, and students need to recognize it as such.
Fermi estimation: The number of bicycles on Earth is probably of the same order of magnitude as the number of people, i.e. in the billions. There are several ways to guesstimate this, but the important point is that any calculation resulting in, say, less than a thousand or more than a trillion bicycles is almost certainly nonsense.
Physical and logical constraints: Little Timmy might own zero, one, two or even three bicycles. He most likely does not own 0.3675 bicycles, and he definitely does not own $\sqrt{-5}$ or infinitely many bicycles. That would just be ridiculous.
Just plain common sense: A bicycle does not travel faster than sound (unless fired from a gun or dropped from orbit). Any answer that implies this is nonsense. Any question that expects such an answer is equally nonsense.
Most children naturally learn these skills at an early age, but unfortunately, through the efforts of all too many early and middle school math teachers, a large fraction of them manage to forget them, or at least to acquire the regrettable belief that such sensible rules only apply outside the classroom. Thus, in order to give such students a practical level of everyday numeracy, and to help them see mathematics as a useful tool rather than just a pointless classroom exercise in memorization and rule-application, these skills need to be re-taught as an explicit part of any responsible math curriculum.
Exams and homework exercises — being, for better or worse, an unavoidable part of most math teaching methods — also need to support these lessons, or at least should not actively undermine them, as "word problems" like those quoted above do. Word problems as such are a great tool for teaching practical problem solving, but only when properly used. As a responsible teacher, you need to make sure that any problems you present have sensible answers, and to instill in your students the understanding that an answer is not complete until it has been verified, and that a nonsensical answer is worse than wrong — a wrong but plausible answer may still be worth partial credit, but a clearly nonsense answer, presented with no indication that the student understands that it must be wrong, is simply unexcusable.
If other teachers are undermining your teaching by giving your students nonsense questions like those quoted above, I'd suggest first having a friendly chat with those teachers about it, perhaps even pointing them to this thread. If that doesn't help, I'd go ahead and tell your students that, while other classes might include word problems that don't make any sense, yours doesn't (or, at least, shouldn't) — and that, even in other classes, they should still make as much use of the answer-checking skills you've taught them as possible. (Make sure to note that, even for nonsense problems, double-checking the answer still helps them catch mistakes in exams. That's always a good motivation.)
As tempting as it may be, though, you may wish to refrain from suggesting to your students that, when they answer word problems from other classes, they should point out any nonsensical assumptions made in the problem statement in their answers. Or maybe not — it depends on the situation.
Addendum: As noted by neminem in the comments, one possible exception to the rule of keeping word problems reasonable are problems deliberately meant to be funny and absurd, and clearly signaled as such. The last part is important — students need to clearly recognize that certain parts of those particular problems may violate common sense. One way of doing so, as noted in neminem's comment, is to pick an inherently funny animal or other being or object, and make clear (via your lecture examples, and/or simply by openly stating so) that, in your exercises, those particular creatures or things may frequently violate laws of physics or common sense. That gives your students an easy way to identify not-so-realistic exercises — "that's another one with a llama, it's probably silly" — while also setting up a running gag.
