Decision-makers are often faced with limited information about the outcomes of their choices. Current formalizations of uncertain choice, such as the explore-exploit dilemma, do not apply well to decisions in which search capacity can be allocated to each option in variable amounts. Such choices confront decision-makers with the need to tradeoff between breadth (allocating a small amount of capacity to each of many options) and depth (focusing capacity on a few options). We formalize the breadth-depth dilemma through a finite sample capacity model. We find that, if capacity is smaller than 4-7 samples, it is optimal to draw one sample per alternative, favoring breadth. However, for larger capacities, a sharp transition is observed, and it becomes best to deeply sample a very small fraction of alternatives, that decreases with the square root of capacity. Thus, ignoring most options, even when capacity is large enough to shallowly sample all of them, reflects a signature of optimal behavior. Our results also provide a rich casuistic for metareasoning in multi-alternative decisions with bounded capacity.