Can El Niño and La Niña occur at the same time?
By definition, El Niño and La Niña cannot occur at the same time. They are instead opposite extreme phases of the El Niño Southern Oscillation.

El Niño conditions occur when the average temperature of the Niño 3.4 region is a half degree or more over the average for three consecutive months or more. La Niña conditions occur when the average temperature is a half degree or more below the average for three consecutive months or more. The reason for looking at a rather narrow band of the equatorial Pacific rather than looking at the Pacific as a whole is because of teleconnections. Whether El Niño, neutral, or La Niña conditions apply has wide ranging impacts across the globe.