Albert Einstein was the major opponent to this view. He could not accept that the statistical nature of quantum theory was the final word. In a paper together with Podolsky and Rosen from 1935, he introduced a modest definition of "an element of reality". It was an attempt to show that certain physical features had to exist even prior to observation, i.e., that they were "real".
The conclusion of this paper was that although such quantities did seem to exist, they were not reflected in the formal structure of quantum theory. Therefore they concluded that quantum theory must be incomplete.
They did, however, opt for one way out of the dilemma, which they immediately turned down. It was that nature is nonlocal, so that a measurement in a certain position can influence observations in other positions.
The latter idea gained momentum in 1964, when John Bell presented an inequality showing that quantum theory could not be both realistic (that physical observables possessed definite values even prior to observation) and local (that a measurement in one position could not influence observations in another position).
Recently, Gisin has shown that any pure, entangled state violates local realism. Work is also under progress indicating that any state which cannot be expressed as a mixture of product states must violate local realism.
Thus it seems rather to be the rule than the exception that quantum systems violate local realism. So why isn't this observed in daily life?
The reason is "decoherence". A macroscopic system posesses very many degrees of freedom. Quantum coherence is preserved only as long as we "keep track" of all these degrees of freedom. If we perform a measurement only on a subsystem of a macroscopic system, quantum coherence (and locality violations) tend to disappear.