A common reason your sprinkler system loses water pressure from a well is simply that there is too much demand being placed on the supply at any one time. A private well pump is designed to deliver a certain amount of water at once. Combining a sprinkler system with normal household demands such as a washing machine, showers and a dishwasher might be more of a load than the well can deliver. The easy answer is to this issue is to only run the sprinkler when other water demands are low.
A leak in the system could definitely cause a drop in water pressure to the sprinkler. You could be losing pressure to a pinhole leak --- or larger --- at almost any point along the way. You might especially be suspicious of this cause if the well, pipes and pump are old. Unless it's an obvious gusher, you might have to hire a professional well technician to try to trace the leak, if that's what the problem is. Even a small leak can cause a drastic drop in water pressure.
A private well works by drilling a hole deep enough into the ground to access the underlying water table, which leaves the water pressure of the system at the mercy of the height of the table. If the water table declines, as it is likely to do during prolonged periods without rain or if a higher than usual demand has recently been placed on it, there's a good chance the water pressure will decrease. The water pressure will likely stay low until the below-ground aquifer has time to replenish itself.
Opening a spigot or turning on a water hose attached to a sprinkler places a corresponding increase in pressure on the well's pump. When the pressure hits 40 pounds per square inch (psi), the pump kicks on and water begins to flow. When the pressure hits 60 psi, the pump turns itself off, and the pressure drops until it hits 40 again, when the cycle starts all over again. With most private well pumps designed to operate in this range, it's natural to experience frequent surges and drops in water pressure brought about by the very nature of the technology.