Can a reaction with positive Gibbs free energy change ever occur?
Yes, $\Delta G = -RT\ln K$.

If a reaction is endothermic with a negative entropy change, $\Delta G$ is positive.

That $\Delta G$ is positive only means that $K&lt;1$.

$\Delta G$ would need to approach infinity for $K$ to approach zero.  

Therefore, at equilibrium, there will always be some products in principle no matter how unfavorable the reaction.  

As a practical matter, the reaction would need to be only slightly endothermic and having only a slight negative entropy change to get a significant amount of product.  

However, if the small amount of product is continuously removed from the system, as for example by precipitation from a liquid phase, or gas bubbling out of a liquid phase, or by liquid-liquid extraction, one can keep the reaction going forward.