As I pointed out in the comments, I think that there are some points in your question that are left fairly vague and to get as concrete an answer as possible would require them to be made very explicit. However, given this problem and taking it at face value I think that we can answer the problem like this:
Let's assume that there are many different subjects, we can call them subject A, B, C, and so on. Each of these subjects has a list of problems that pertain to them, and this list might be infinite. Furthermore, we are able to categorize each of these problems by the skill level of somebody who wants to solve them. The levels start at 0, where anybody at all could solve them, and then move upwards in increments of 1 (0, 1, 2, 3, 4, ...).
If we know of some person Alice and are told that she has a skill level of L in subject A, that means that she would be able to solve any problem in A that is of the skill level L or lower. This means she can solve the problems in levels L, L-1, L-2, L-3, and so on until we get to 0. As an example, suppose she has skill level 5 in subject A, she would be able to solve any problem that requires a skill level of 5, 4, 3, 2, 1, or 0. So far, this all seems to make sense just based off the idea that we want to quantify someone's ability to solve problems in a specific subject while those problems are also quantified in terms of how hard they are to solve. If we only quantify someone's skill and don't quantify specific problems, I have no idea how you could come up with an answer to that question (how else could you quantify skill level?).
Will there always be a problem that she cannot solve? That depends on two things. First, is there an infinite amount of levels of difficultly with at least one problem at each difficulty in any subject? By that I mean, for example, does A have levels 0, 1, 2, 3, ... to infinity with at least one problem? Secondly, can Alice reach higher levels of skill at a subject if she practices that subject? If there are infinite levels to a subject with infinite problems then Alice will never be able to reach the top, there will always be one more above her that she cannot do yet. That's assuming, of course, she can increase her skill with practice. If Alice cannot increase her skill and there is a subject, B, with at least one level higher, L+1, question than her level, L, in that subject, then she will not ever be able to answer that question because she can never get to a skill level of L+1.
So, in answer to the question "could we give her a sequence of problems that would trip her up?" Well, again if we are defining everything in the way above, we don't need to give her a sequence. We only need to give her one question that is L+1 in a subject where her skill is L: give her a 6 when she is at 5. If you want to give her a sequence of problems that are level L, a sequence of problems that are level 5 in our example, then of course she will be able to answer all of those problems, she has a skill level of 5 and they are all level 5.
However, this is where things get very vague and I think a more specific question is required to understand what is going on. If you are asking about how this works in practice, then sure anybody would eventually get tired of doing problems over and over again, let's say math problems, and eventually they'd get one wrong that they could easily do if they weren't tired. Everyone messes up basic arithmetic every once in a while, especially when they're tired. But again, this is subject specific. Are we talking about math? Science? History? Philosophy? Engineering? There isn't an objective answer to the question "can we give her a sequence of problems that will eventually trip her up" unless we know 1) what kind of problems these are, 2) what subject we're talking about, and 3) are we talking about a real person or some idealized person who has a perfect ability to do problems that are at their skill level. A perfect Alice would never mess up a level L problem, no matter how many we give her. A real life Alice would probably mess up a 0 level problem every once in a while even if she's level 5. It depends on myriad factors about the situation and for that type of answer we need to have a more specific question.
This goes double for the question "is there a subject where she could lie about an answer and get away with it?" That completely depends on what these subjects are. Are you asking about the entire list of anything that can fall under the title of "subject"? That would require the entire list of subjects and I don't think any person has such a list. That question is entirely dependent on the specifics of what you're talking about, the domain of discourse we could call it, and it seems that the domain is fairly vague right now.
On the other hand, if we assume some things and take a mechanical approach to this like we did above, then as long as there is someone who can do a problem at level Q, Alice will never be able to fake a question at level Q because that person could just do the problem themselves and see if she got the right answer. Could Alice lie and convince this person? I do not know, that depends, again, on a myriad of factors like "are these idealized people or real people", "do they have completely rational thoughts", "do they ever make mistakes", and so on.
Hopefully this helps clarify any of the problems you were thinking about, and as was talked about in the comments to your question, if you are inspired to reformulate a new question into a more specific problem there will undoubtedly be a more specific answer.
