Is it ever reasonable to come across an event that is very improbable and then infer that it must have been designed?
In general, "no". In order for a proposed cause to be considered viable, it should also possess sufficiency.
That said, "sufficiency" tends to cut against non-design causations. For example, random processes have never been shown to produce meaningful information (e.g. the works of Shakespeare, or a functional biological organ) from "scratch". Conversely, intelligent agents manipulate the world in a way to actualize extremely improbable outcomes all the time, so while the answer here is technically "no", it's unclear what situation could arise in which an intelligent agent (of suitable capabilities) would not be a sufficient cause for some improbable event.
For example, getting three royal flushes in a row is incredibly improbable.
Here we see the need to introduce another factor: specificity. Getting any specific three hands of cards is extremely improbably. Nevertheless, if I sit for three hands at a poker table, an extremely probably event will be actualized. "Entropy" plays a role, but what we're really trying to measure is "randomness". This is a whole, genuine field of study that can be qualified, but to keep things simple, "interesting" outcomes are both specified and complex.
A deck of cards adds some complications, so consider, instead, a series of coin flips. The outcome of all-heads is highly specified, but not complex; the most likely explanation is neither chance nor design, but necessity (namely, your coin probably isn't fair). A series of perfectly alternating heads and tails is more interesting, but still not complex enough to strongly suspect design rather than necessity. (For a more "real world" example, crystal growth is an example of necessity.) If, on the other hand, there is no discernible pattern, then the outcome is highly complex but is not specified, and is likely a result of chance.
Now, if you flipped a coin, and the flips, when "read" as binary representation of ASCII, spelled out the Declaration of Independence, you're dealing with a result that is both complex and highly specified; design is now a much more likely explanation than chance. Similarly, any three "typical" hands of poker are extremely improbable, but they are not specified. Three royal flushes are both unlikely and specified, therefore making design (cheating) a more likely cause.
Now... here's where things get interesting, because this is the point at which the infinite monkey theorem is usually invoked (including by several existing Answers here!). While technically correct, the theorem only works in the world of infinities, and only in certain conditions. To understand, however, it is necessary to clarify the nature of our unusual event.
Going back to the poker example, if Bob sits down and is immediately dealt three royal flushes, that's cause to suspect chicanery. If, on the other hand, Bob has been playing poker for 1016 hands, then it is much less "surprising" if, at some point, he was dealt three royal flushes in a row.
I'll circle back to this in a moment, but it's leading us to another point:
Is there a probability X such that if the event's probability was below that X, we can safely conclude design?
Yes. Often called the "universal probability bound", if we estimate the total number of particle interactions that have occurred in the history of the universe (usually cited as around 10150), then anything unlikely to occur given that many "chances" is unlikely to have occurred by chance. (Obviously, the number of poker hands that can be played is much, much lower than this number. More "realistic" estimates use a bound of 10-50 as the threshold for excluding chance as a plausible explanation.)
The oft-cited solution is to posit that existence is, in fact, infinite (a.k.a. the anthropic principle). The problem with this, besides being a wholly philosophical supposition with no concrete evidence to support it, is that it doesn't actually work in practice... and this is where we come back to the monkeys and how theorems about infinity don't work in a finite universe.
Let's use another Answer's Dyson sphere as an example. It's theoretically possible (albeit far, far below the universal probability bound) for such a structure to spontaneously materialize... just as infinite monkeys will eventually produce the complete works of Shakespeare. However, in an infinite universe, there will be infinitely many humans that have not encountered a Dyson sphere. The probability that, by chance, our universe happens to have both humans and Dyson spheres is itself unreasonable. In other words, the anthropic principle cannot defend chance as an explanation. The only plausible explanation for our universe happening to have both humans and Dyson spheres is either design or necessity.
Now, I can imagine no argument that makes the existence of Dyson spheres necessary in order for there to also exist humans to wonder about them. Moreover, it's difficult to imagine even the many unlikely forms of life that we do observe being necessary for observers to wonder about them.
To be fair, technically the humans in the universe with spontaneously generated Dyson spheres could still fall afoul of the anthropic principle, but this requires belief in infinite universes, for which we have no scientific evidence. Such belief is purely philosophical. Worse, because the probability of Advanced Aliens existing in an infinite multiverse is non-zero, it's actually more likely that a universe contains Dyson spheres because someone built them. In fact, if the probability of Advanced Aliens being able to create entire universes is non-zero, it immediately becomes more probably that any "interesting" universe is, in fact, designed.