In 1995 Wolfgang Ketterle at MIT produced a Bose-Einstein Condensate in a gas of sodium-23 atoms, but sodium-23 doesn't have an integer spin. How does this work?
The protons, neutrons, and electrons that make up any atom are all Fermions with spin $1/2$. You can't make a Bose-Einstein condensate out of just electrons, for instance. But a composite particle can have a net integer spin.
As an example, a helium-4 nucleus has two protons and two neutrons. The nucleons are arranged according to the nuclear shell model. Both the neutrons and protons are paired resulting in a net spin of zero. In a helium atom there are two additional electrons, which are also paired, so the whole neutral helium atom has a net spin of zero. A helium-4 atom is a composite boson.
Sodium-23 has nuclear spin of $3/2$, making it a fermion. There are 12 paired neutrons, 10 paired protons, and one leftover unpaired proton. The unpaired proton sits in a shell state which contributes the spin of $3/2$. But the Bose-Einstein condensate is formed by atomic sodium, not nuclear sodium. The 11 electrons in a neutral sodium atom contribute an unpaired spin of $1/2$ to the total. The full sodium atom is a composite boson, so it can form a Bose-Einstein condensate.