The FAA (Air Traffic Control (ATC)), not DIA, has sole authority and responsibility for routing and separating aircraft throughout the national airspace system. ATC's first priority is always the safe and efficient separation and routing of aircraft throughout the national airspace system.
The Denver region experiences many types of aircraft over flights that may or may not be related to DIA. In the Denver airspace, at any point in time, there could be long haul flights en route (e.g. from Los Angeles to Chicago) traveling at very high altitudes; general aviation aircraft traveling to and from other local airports, military aircraft, other commercial aircraft traveling to nearby commercial airports and helicopter traffic for medical or traffic reasons.
If you live anywhere within the Denver metro area, you will likely experience aircraft over flights. How and to what frequency depends on the weather, the runways being used, the type of aircraft, aircraft engine characteristics and relative distance from the airport.