Obviously we can't see the Big Bang, and we can't make another Big Bang in our lab (ignore anyone who tells you that the LHC recreates the conditions of the Big Bang, because it doesn't). So what we do is to devise theories to explain why the universe looks the way it does, and then we do experiments to compare our theory with the real world. If the experiments match our predictions then we feel confident that our theory is correct.
In this case the theory we use to describe the universe is General Relativity. When we make a few simplifying assumptions about the universe and feed these into General Relativity it tells us that our universe is described by an equation called the FLRW metric. We've used this to make predictions about the universe, and so far they've all been proved true, so we think the FLRW metric really works.
The key prediction of our theory of the universe is the cosmic microwave background. The WMAP satellite has measured this to extreme accuracy and it perfectly matches what we'd expect.
The thing about the FLRW metric is that we can use it to wind time back and calculate how the universe must have been in the past. If we do this we find the universe gets denser and denser and hotter and hotter, until around 13.7 billion years ago the density and temperature become infinite. This is the point we've called the Big Bang.
The problem is that we can't do calculations when physical properties like density and temperature become infinite. That's because infinity isn't a number and you can't feed it into an equation. This means no-one knows what actually happened at the Big Bang. But we can get very close to it. In fact we can get to within 1 picosecond, that's $10^{-12}$ seconds after the Big Bang. That's how back the LHC takes us - to get any further back we'd need an even bigger accelerator, which i guess we'll build some day.
So when you say:
a point arrives where the scientists give up
well that's quite correct, and there's an excellent chance we'll never know exactly what happened at the moment of the Big Bang itself. However if we can calculate back to 1 picosecond after the Big Bang and get the right answers then you'd have to concede that if our theories tell us that the Big Bang happened then it probably did.
Vote to close as not a real question.
– AdamRedwine May 25 '12 at 18:57