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Yes, it is possible, but it is extremely rare. There are only certain STDs that can be transmitted by blood transfusions, but all donated blood is tested for these germs.
The current tests for the STDs that can be transmitted by blood and body fluids are very accurate and will almost always detect the STD germs that can be transmitted by blood transfusions. However, there is a very small window after which a person is infected that the tests are not yet positive.
The current estimate of risk of transmission of HIV via a blood transfusion is 1 in 2 million. The risk of hepatitis C is similarly 1 in 2 million. The risk of hepatitis B is higher, with a risk of 1 in 205,000. The National Heart, Lung, and Blood Institute notes that the risk of acquiring HIV via a blood transfusion is lower than the risk of being struck by lightning.
The risk is so small that it is not recommended that anyone should be tested for these germs unless they develop symptoms that are worrisome for the respective infections.
If I can provide any additional information, please let me know.