Yes, the statement in the book is sensible.
Consider a physical system with two possible "states", like: Electron spin = pointing up or pointing down, or light polarization = clockwise/anti-clockwise (with respect to it's direction of propagation).
Examples like door = open/closed and schrodinger's cat = dead/alive make it sound totally weird since we never experience such in our daily experience. So I would prefer to avoid them. There are reasons why such quantum effects are observed only on tiny scales, and not in day to day classical phenomena.
Intuition from day-to-day life suggests that the electron spin can be either up, or down, but not both. But that's an incorrect picture to describe the system. It is a better description to say that the electron is in a "superposition" of the two states... like say $0.8|up\rangle + 0.6|down\rangle$. Note that the coefficients don't sum to one. Instead, the squares of coefficients sum to 1. So these coefficients are a little different from usual probabilities -- they are called probability amplitudes. So not that here, the electron is in a state which has both P=up (partly) and (~P = NOT up) partly. Genralizing from this example, states that are completely P (or ~P) are edge cases. A generic state will be a combination of the two possibilities. Now you can understand what the author means by that statement. Classically you would have said that the electron spin must have been up or down, but not both based on the probability rule prob(up) +
prob(not up) = 1. But that's not true any more and the system simultaneously is both up and down, in a specific way.
Important note : This does NOT mean that quantum mechanics is illogical. In fact, this means that the usual rules of probability are too crude and not subtle enough to describe the behaviour of physical systems on small scales. We have developed/found other mathematical structures to precisely and accurately describe quantum phenomena.
An electron can remain in this state till it's "measured". The best description we have so far is that as soon as you measure whether such a system is spin up or spin down, the electron immediately falls into one of the two states! (It's like the electron doesn't want to be caught with it's pants down and one foot in each state, but it also can't make up it's mind till it's forced to do so by someone measuring it. And don't take this explanation too seriously.)