It is possible for a patient in a modern hospital to have a ct of the brain (non-contrast) & a chest x-ray only 3 minutes later? this seems very fast.

Yes; Radiology Dept.: Modern hospitals have all the Radiology equipment concentrated in 1 department. If the doctor orders a battery of tests, it is possible to have the patient transported from the CT scanner to an X-ray room next door. Also, if they are prepared, the X-ray can be taken as soon as they arrive. Portable X-ray equipment can also be used in the waiting area, though the quality is worse. HealthTap Prime.

In an extreme: emergency situation, in which the X-ray room or equipment is held open and waiting for the patient to finish the CT, yes. Under normal circumstances, even if the X-ray room was right next door to the CT scanner, a 3 minute turnaround would probably not happen unless there was a conscious effort to expedite above and beyond the normal workflow