Is it possible for a person to have both sickle cell anaemia and thalassemia at the same time?

Yes: You can be a carrier of both.

Yes: The mutations causing sickle cell disease and beta thalassemias both occur in the beta hemoglobin gene. There are many people who have one copy of beta hemoglobin with the sickle cell mutation, and the other copy carries a beta-thalassemia mutation. Depending on the combination, symptoms can vary. For a nice summary, see: http://www.Cdc.Gov/ncbddd/sicklecell/facts.Html.

Yes!: Sickle cell and beta thalassemia both affect the same part of hemoglobin. If you inherit a sickle gene from one parent and a thalassemia gene from the other, you have sickle/b-thalassemia. The severity depends on whether the thal gene works a little (b+) or not at all (b0). Sickle cell and Alpha thalassemia are inherited independently; you can inherit both from the same parent or one from each.