By normally you mean 8,000 feet?
This is an interesting topic that has been looked at quite a bit given the number of people with cardiac and pulmonary disorders. (This is by no means a complete list, just a couple that get used)
Kelly et. al. Directly measured cabin pressure conditions during Boeing 747-400 commercial aircraft flights. Respirology. 2007 Jul;12(4):511-5. PubMed ID: 17587417
Study recorded cabin pressures and all flights fell within 8,000 foot limit set by the FAA. 45 flights one aircraft and three airlines.
Cotterll. Altitude exposures during aircraft flight. Flying higher. Chest. 1988 Jan;93(1):81-4. 3335171
Study showed a wide range with a mean cabin pressure of 6,214 feet (1894 m) and a max cabin pressure of 8,915 feet (2717 m). 204 flights on 16 different aircraft, and 28 airlines.
The 8,000 foot limit is an FAA limit and last I heard, airlines in international air space could make their own call as to pressure to hold (not a pilot, just a researcher so PLEASE correct me here). I know there was an abstract at a Aerospace Medical Association meeting a few years back that tracked trans-atlantic flights but I can't find it.