We study the single item lot-sizing problem with supplier selection and uncertain lead time. We consider the situation where a company had preselected a set of suppliers for an item, and these suppliers have different prices, different lead times, but also different reliability. We aim to provide a robust optimization approach to decide when to order, how much to order, and from which suppliers, in the context of un- certain delivery lead time. We formulate the robust optimization prob- lem with polyhedral budgeted uncertainty sets. This formulation does not require assumptions on order crossovers, order splitting, or on the structure of the demand or lead times. We propose an exact row and column generation algorithm to solve the considered problem, along with some enhancements including a fast cut generation procedure. To improve the scalability of the approach we propose several heuristics, including a hybrid of the robust counterpart reformulation and row and column generation, and a fix-and-optimize approach in the row and column generation framework. Experimental results show that the fix- and-optimize approach provides good results. Finally, we provide in- sight into the reaction of the decision-maker to unreliable suppliers. One of the conclusions is that in the considered framework, an ex- tremely risk-averse decision-maker selects a single supplier, namely the most reliable one even if it does not offer the lowest price.