No, it can't.
Ultimately, this fact is decided by observation - but it fits within a general property of the laws of physics that is known even from circumstances closer to everyday human scale - that motion is relative.
What this statement means is the following. If you consider a particular object and its surroundings, the laws of physics will generate the same future history(*) in the situation where that object is moving and the surroundings are stationary, as where the object is stationary and the surroundings are moving in the complementary way (i.e. with the opposite direction and speed), provided of course we also likewise adjust the future-generated history from the first scenario to bias motion accordingly to translate it into the second one.
Suppose that there existed a particle with the property that it could spontaneously burst, like you imagine, at a suitably high speed. Say that when it gets suitably close to the speed of light (about 300 megameters a second [Mm/s]), it becomes unstable and bursts after a short flight time, say 4 seconds. (Note: for anyone who knows about real particle decays - which I'll touch on later - this is obviously "too deterministic" but I'm trying to keep things simple.)
Now imagine what the history might look like: at second 1 it travels 300 Mm, at second 2 it has traveled 600 Mm, at second 3 it has traveled 900 Mm, then at second 4, just as its odometer finally hits ~1200 Mm (about 3 times further than the Moon!) it suddenly bursts into a shower of quarks.
Now take that this same behavior was not observed were it "staying home". 10 seconds, 100, 1000 later it would still be there, whereas the moving particles would reliably burst at 4 seconds and ~1200 Mm displacement once at or above, say, 99.99% of the speed of light.
Now consider two such particles of the same type, starting out from the same point. One is "resting", the other is moving "near the speed of light". The second particle bursts at 4 s in, while the first remains integrate forever. Remember what I said above about transformation. Suppose I transform this history so that the "resting" particle is now moving backwards at 300 Mm/s, and the "mover" particle is resting. If the motion-is-relative property is to be upheld, then we must have one of two things:
if the "particle decays in 4 seconds" property is to be upheld, then in this transformed history we must have that the now-resting "mover" particle bursts at some point. But resting particles cannot burst,
if the "particle is stable" property is to be upheld, then the now-moving "at home" particle, 100, 1000 s, or even many ages of the universe, later will still be intact, in contradiction to the rule that "near light speed" particles must burst after 4 s.
Or to think of it another way, simply shifting the motions of things mathematically cannot create or remove a complex phenomenon like the rupture of an object into a shower of smaller objects. Thus, if such a thing did indeed occur at one speed but not another, we can use that to infer our speed: we just have a particle of the given type at rest with us, and if we see it burst in front of our eyes, then we know we must be travelling at 99.99% of the speed of light - absolutely, and with enough measurements of this phenomenon we can extrapolate a universal reference frame with regard to which we can say everything is .
Ultimately though, as mentioned, the undetectability of any putative universal reference frame in this form is still only an empirical truth. Maybe there are such particles out there - but we haven't seen one, and it gets less and less likely the more evidence we get, just as with anything in science. And likewise with scientific reasoning, the strong track record of the relative-motion principle lets us say that
All the above said, there is an important subtlety here that deserves mention. We know of many particles that do spontaneously "burst" as just described: they are called unstable particles and, in fact, most of the particles on the Standard Model are unstable as are most composites of quarks except the proton, and the neutron when bound up in a nucleus (again, as far as we can observe!). But there is also something we can observe with them that may make you question this: if we take, say, a muon - essentially a heavy version of an electron that the Standard Model provides - which "at rest" decays in an average of 2.2 μs, then we boost it to near the speed of light, we will see it decay slower than that - not faster, but slower: e.g. suitably close it will now take ~2200 μs, say. Does this mean speed "stabilizes" particles?
The answer, again, is no: first, keep in mind that in one case we have no decay at all versus the presence of a decay, whereas here we only have a shift in the amount of time. But second, and more comprehensively, we can see that this can be fully accounted for by only slightly more complicating the transformation that we use when relating histories at rest versus motion - i.e. we need to do a few more adjustments to convert one history to another as in what I described prior before the laws of physics will "accept" it as truthful, but they are still relatively simple and, more importantly, universally applicable mathematical transformations, meaning that even in completely different situations the same transformation will still preserve the same physics. This a basis of special relativity, but the point is the laws of physics still have a motion symmetry, just "shaped" a little different.
(*) If you prefer, another way to think of laws of physics is that they "verify" particular candidate histories as being "physically plausible" or not, and we are saying that if we change each history suitably into the other by a regular mathematical transformation, the laws of physics will once again verify it as valid.