Something might be thermodynamically unstable so that its products are lower in energy than the starting materials but the reaction might be so immeasurably slow that from our point of view it's effectively not taking place at all. Or the activation energy might be so high that the reaction won't occur without an initial input of energy to give it a 'kick-start'. If this is the case then we say that the substance we are considering is thermodynamically unstable, but kinetically stable. If the activation energy is low enough so that the reaction does occur at a measurable rate then the starting material is both thermodynamically unstable with respect to its products and kinetically unstable.
If you would like an example which explains why this might occur (if not, then ignore this paragraph), the example that was given to me in my chemistry course at uni was the sulphate ion (SO4)2- Vs the sulphite ion (SO3)2-. Theoretically, both ions could undergo reactions which would result in thermodynamically more stable products, but the geometry of (SO4)2- gives it kinetic stability. (SO3)2- is a trigonal pyramidal shape with a lone-pair on a the open face, meaning the ion is susceptible to electrophilic attack, whereas (SO4)2- is a closed, tetrahedral structure (i.e. you would have to break the bonds first) with no lone-pairs, making attack from another species difficult, so it is fairly inert and therefore kinetically stable.
Usually in a question regarding energy graphs, you'll be given an exothermic reaction where the products are at a lower energy than the starting materials. As you probably know, the line showing the energy has a small hump which goes up (showing the activation energy) before it drops down sharply to the energy level of the products. When the starting materials are kinetically stable, this initial hump will be relatively large, showing the high activation energy barrier which must be overcome first in order for the reaction to proceed.
If by some chance you were given an endothermic reaction in a question, this issue would not exist as endothermic reactions don't really have an activation energy as such as the products are higher in energy than the starting materials anyway.