Some municipalities have what is called a combined sewer. In a combined sewer, all or some of the water that is drained from storm drains flows in to the same sewer system where waste water from homes is deposited.
If you have this type of system in your area, a basement drain that's backing up may be a sign of an overflowing sewer system, a problem much larger than just your basement drains.
If, either due to damage or improper installation, a basement drain is not tied in to a drain system, a rising water table due to prolonged rain can cause backups through a basement drain. In some older homes, basement floor drains were little more than open slots that allowed water in the basement to flow down in to the soil under the house.
During periods of heavy rain, the opposite can occur, with water from the soil flowing into the basement.
Tree roots can easily damage and even collapse underground drainage systems. It is not uncommon for a tree that is seeking water to break open old iron or clay pipes, and cause a back up of the system.
The tree roots clog the drain, and the drain in turn backs up and overflows when it rains. The long term solution is to replace the pipe, but the short term solution can be a simple snaking by a plumber with a professional grade plumber's snake.
Rising Water Table
The rising water table in an area experiencing prolonged rain can cause basement flooding. The first place water begins to pool is the lowest parts of the basement floor, which are typically where any drains would be installed.
If the drains are tied to the municipal sewer system, and that system is full from the rain, the perception can be that the drains are overflowing. The truth, in this case, is that they are merely not draining.
For these sorts of scenarios a basement sump pump drained well away from the house is recommended.