The problem here is that there are
two distinct areas of physics. The classical one and the quantum mechanical one.
What
goes on in the atom is essentially quantum mechanical. The laws of classical physics simply
do not fully explain what is going on there.
Think of the electron in orbit, in a classical
situation it is accelerating and will therefore radiate energy. However Bohr's theory forbids
this unless there is a free space in an energy level below it – as you know it cannot simply
spiral inwards.
However in the quantum mechanical world we should think of it as being
in a static stable state and therefore the word acceleration has no real meaning. Also if you
consider the electron to be a wave function then any concept of acceleration is again not
relevant.
So to summarise there is nothing wrong with the classical theory, in many situations, it is
just
that it does not apply here.