This is a particularly sticky wicket because (a) we are delving into the philosophy of quantum mechanics (which is beset on all sides by those who wish to pervert it to their own ends) and (b) because we are confounding our understanding of the problem by confusing what we mean by "exists."
To make this question more meaningfully answerable I will address precisely what is meant by the idea that "nothing exists until we measure it" and then go on to provide an answer to this newly reformulated question.
Who can really know what Bohr was thinking when he made the statement that you quote in your question, but what I assume that he meant is not that the "particle" (as one example) does not exist before you "measure" it but rather that information about the particle's "observable" attributes (such as position, momentum, etc.) does not exist before that observable is measured. In other words, quantum theory claims that all the information that we have about a particle is contained in its wave function, which generally represents a linear superposition of possible states in which the particle might be found if a measurement is performed.
Therefore, one might ask, "What is going on before we make a measurement?"
Historically, there were three answers to this question:
(1) Something totally predictable is happening, but quantum mechanics doesn't know what it is because quantum mechanics is incomplete. (The "Hidden Variable" Answer)
(2) Maybe something totally predictable is happening, but we'll never be able to know what, so don't even bother asking.
(3) Whatever is happening looks just like quantum mechanics and therefore doesn't affect the outcomes of our experiments, so who cares?
As it turns out, the first answer is basically dead because of Bell's Theorem, which essentially says, "If the predictions of quantum mechanics are correct, then local hidden variable theories do not describe reality." In other words, no theory which allows us to know exactly what the particle is doing at every moment in time (and also preserves causality) can ever reproduce all of the predictions of quantum mechanics.
The second answer is demonstrably false because a theory can be constructed which tells us exactly what the particle is doing at every moment in time by violating causality. It is known as the "pilot-wave theory." The theory exactly reproduces the predictions of quantum mechanics but is widely disregarded as a "toy theory" because it doesn't make any new predictions of its own.
As a result, answer #3 in conjunction with the Copenhagen Interpretation tends to be the default position for most physicists.
The take-home message is this: if you insist on things being "classical" (i.e. having definite observable values) to exist, then you must sacrifice local causality. On the other hand, if you are willing to accept the existence of things that don't have definite values for their observables (that is, accept the idea that an electron is its wave-function - whatever that means), then things certainly exist before they are measured. However, just as easily, you could say that "nothing exists" before a measurement and the act of measurement forces the system to "take a stand."
Any way you cut it, there is definitely one thing that you can't say: namely, that the system has some definite value but we just don't know what it is. Either it has a definite value and we know it, or it doesn't and we don't. You can have one or the other but not half-and-half.