It's really not the limits of science that pose the problem here. It's the limits of eyewitness accounts and subjective, unrepeatable experiences.
Scientific experiment and mathematical modelling are also subjective experiences. But they are subjective experiences that can be repeated again and again (and witnessed by many others who also understand how to perform the same experiments...).
OK, this is obviously what you believe, and you also sound like a skeptic (the type that doesn't actually study the subject, just debunks it).
Science and math absolutely do not guarantee repeatability. The same experiment has often been done by multiple groups with differing outcomes. Especially when it comes to quantum physics. Same thing with math. They can make the math work for anything, but it gets ugly. When you get into quantum mechanics, the math and a lot of other things break down. They can't fit gravity into any of the equations. That seems like a real problem, doesn't it? Once they figure out how to do that, I'd bet the rest of the theories will break, and then they will have to start over again! It wouldn't be the first time.
This is where all the M-theory and dark matter/energy stuff comes in. They need them to fix holes in the math. But the standard thinking wont prove the ideas, so it's theoretical physics. How do you prove other dimensions exists if you can never detect them? So they have to try new stuff, and in the end some answers will never be known, not at the quantum level anyway. Scientists are prevented from seeing certain things.
We can say that we can't break the light speed barrier, and then they end up doing it in the lab, and that's not even counting quantum entanglement, or what Einstein called "spooky action-at-a-distance".
Science is often educated guesses. Then they try to "prove" the hypothesis with experiments, hoping to end up with a theory. But that doesn't always work, unless you are talking about materialistic science.
Go back and listen to the August 26, 2007 show with T. Allen Greenfield and Jeff Ritzmann. They talk about the limitations of current science. Great show!
One last thing to think about is when it comes to UFOs.... no one is going to prove anything anytime soon. We know that thousands of people see these things. We know what they aren't. We can argue if a photograph or video is real or not, and that still doesn't prove or disprove UFOs.
In the end people will continue to see them, and some of those people are very credible and will include pilots, astronauts, and even scientists. Some might even take photos... and if they don't, it doesn't take away from the fact that they saw them.
So far current science can't prove everything.