Yes, it is possible, under some circumstances. However, your general understanding that only one aircraft is allowed on a runway at one time is correct most of the time.
There are a few obvious cases, also covered in the other answers. For instance, an aircraft can be allowed to cross or taxi on a runway behind another aircraft which is taking off or landing. An aircraft waiting to depart can also be cleared to enter/line up on a runway behind another aircraft taking off or landing. These examples are pretty simple, because only one aircraft is actually using the runway for takeoff or landing, the other is just taxiing behind. But actually, two aircraft can takeoff or land on the same runway at the same time in accordance with the below, and as stated in ICAO Doc 4444 section 7.10 on Reduced Runway Separation.
An aircraft can land on a runway while another aircraft in front of it is also landing, provided that the first aircraft has touched down, is in motion and will vacate the runway without backtracking (doing a 180 on the runway).
An aircraft can land on a runway while another aircraft in front of it is taking off from that runway, provided that the first aircraft is airborne.
An aircraft can take off from a runway while another aircraft in front of it is also taking off, provided that the first aircraft is airborne (but still over the runway, and thus not clear of the runway yet).
ICAO suggests some pretty strict requirements in order to use these rules, such as the exact distance needed between the aircraft involved, as well as the type of aircraft, runway length etc. However, some states have pretty lenient rules. In Denmark, for example, it is largely up to the air traffic controller on duty to decide if conditions are safe to use reduced runway separation. The phraseology used is a little different as well:
Instead of the usual "cleared to land", the phraseology used is "Land behind preceding landing (aircraft type)". The second aircraft will obviously have been informed of the first one, and weather conditions must permit the pilot to see the first aircraft at all times.
Similar to 1, the phraseology is "Land behind preceding departing (aircraft type)".
A normal "cleared for takeoff" is used, but the pilot of the second aircraft must have been informed of the preceding aircraft and have it visually in sight.