In short: No.
Compared to 16,000 RPM, 20,000 RPM is 25% over the specified maximum. Manufacturers -- even cheapish ones -- always calculate in a bit of tolerance, but 25% over limit is a lot. I wouldn't mind running a 19,000 RPM bit (if that existed) at 20,000 if it was the only one I had available, but surely not a 16,000 one.
What may happen?
1/16'' (1.5mm) is pretty slim, these bits are not unlikely to break even when treated properly. Most reputable manufacturers here do not even sell such small bits except with very slim shafts (3mm), which are usually only available for "toy" routers (like a 175W Dremel or Proxxon) that are a lot weaker and usually allow speeds as low as 7,000, too. A "typical" router for a Real Scotsman normally cannot fit such slim shafts at all, and for good reason -- you can already predict with high likelihood what would happen if you clamped that tiny fragile bit into the router.
Indeed, when you do find such ultra-small bits here, you usually get them in packaging units of 10, which is tell-tale for their destiny. Or, to cite Robert Louis Stevenson:
"Prophetic," said the doctor, touching this picture with his finger. "And now, Master Billy Bones, if that be your name, we'll have a look at the colour of your blood."
What will happen?
Nobody knows! The two most likely things to happen will be either nothing at all ("works fine"), or the bit ripping off right away near the shaft as you plunge in.
It is usually much more serious to have a wide-diameter bit go off (that's more like standing next to an exploding hand grenade). But of course, even a slim bit may just as well go flying somewhere, including the direction of your stomach or your face. You have no way of telling whether it will just work fine or whether you will die from injury.
Of course, desastrous failure can always happen with any kind of bit, all the time, and for a wide variety of reasons. Among them pushing forward too hastily, tilting, buying low quality bits, not maintaining bits properly, or just... bad luck. And, of course, running 25% over the specified maximum. Life is dangerous and often lethal, and power tools -- routers in particular -- are dangerous beasts.
But there is reasonable risk and unreasonable risk. Reasonable risk is wearing a knife-resisting vest (I'm actually doing that) which will catch 99.9% of all sharp objects that aim for your heart and your liver, and a polycarbonate face mask while working with well-maintained tools at the indicated speed. Or, more traditional, protective glasses and a thick cow leather apron.
Unreasonable risk is... everything else.
What may also happen?
The bad thing about taking unreasonable risk (read as: being grossly negligent) is that not only is desaster a lot more likely to happen, but also that you are out alone in the dark with the consequences.
If the bit sends out shrapnels and you lose an eye or are otherwise injured in a non-trivial way, you may find that your casualty insurance tells you: "Why! You ran 25% over the specified maximum, this had to happen", and refuses to pay the bills. The same can (almost certainly will) happen with your indemnity insurance in case you injure a bystander.
If nothing else, that alone is a good reason not to go too much over the limits. The risks simply are not acceptable.