Yes, depends on the oil, doesn't matter how pure the oil is some EOs break down faster than others.
Seebs is right, citrus oils are the biggest concern, they oxidise faster than any of the other oils to form terpenes as skin irritant. They say they have a one year shelf life, but I like to use them up in 6 months. Also depends on the conditions they are stored in (even temp, lid on tightly). A half empty bottle of bitter orange with it's lid off stored in sunlight will oxidise faster than a full one in a cool cupboard with it's lid well sealed.
Wood EOs will usually last the longest - they're usually part of the tree's natural preservatives, so you would expect them to last longer.
So I ignore the expiry date on most wood and resin EOs, and am VERY cautious with citrus EOs. Other EOs, you can usually notice changes (colour/viscosity/aroma/etc) that will indicate they are have changed and may not have the same properties as the original bottle.