Could scientists test a theory of everything?

A common misunderstanding about theories in general is that you can prove them, when in fact you will never be able to prove a theory - you can only ever falsify a theory. There is simply no way to prove that a theory will always apply to every case that it's supposed to be usable for. You can only ever falsify a theory and thereby say that it doesn't work for a specific case and thereby say that it can't possibly encompass all the cases it was supposed to be applicable for, because you have at least one example for which it doesn't work.

This means that no scientist will ever be able to prove that the theory of everything is correct.

They can only conduct experiments to falsify it. And if they can't falsify it then it's good enough to be used until a case comes up where it's falsified, which would mean that they would need to search for a better theory that is also applicable to that case.

As was noted in the comments this is not a complete definition of what a theory is. For example to be a theory you need to be able to check it. The easier it is to theoretically falsify it the better the theory. The logic behind this is that if there are many points you could attack and anyone could attack the theory at any point without a lot of resources then someone will surely be able to find flaws in your system at some point. If your theory still manages to stand and not be falsified despite experiments being easy to do and many experiments being conducted your theory seems to be usable and people will start to accept the theory as a basis for their work.

High attack surface + after long time still not falsified = good theory

This still means that your confidence in the theory is the only thing that can rise and you will never be 100% sure that your theory is correct. You simply can't be sure that a theory is correct. You can only say that it worked for all tested cases and until a case comes up that falsifies the theory you simply assume that it works this way to make your life easier and continue your work.

If there is no way to falsify a theory then you are in the range of pseudo-science and disregard the normal scientific process. A theory that you can't possibly disprove is by definition not a theory. The same applies to arguments like "There was something here that made it work when I did this experiment, but now it's gone and you can't reproduce it, but my experiment was successful so my results are correct." If another person can't reproduce it under the same circumstances it's useless and not done with the necessary scientific rigor. Yes, the experiment could be very costly or difficult, but it has to be possible to repeat an experiment.