Quantum: why linear combination of vectors (superposition) is described as "both at the same time"?

I want to get a better understanding of quantum phenomena and out world in general. Before long I've thought of Schrödinger cat as being both alive and dead (or spin both up and down). Now after some reading of math, it looks to me it is not technically correct, as the cat/the spin is in a linear combination of vectors in Hilbert space (which is just another vector in that space). In my understanding linear combination of not the same as being both base vectors at the same time and the "naive" viewpoint of the cat being both alive and dead is oversimplification and misleading. Why is it commonly called "both at the same time"?