Can you have an STD and it not be detected?

Yes, sometimes even tests can not always detect an STI.

For example: Lesions that are suspected to be Herpes can be swabbed and tested but may not come back as Herpes. Sometimes a blood test can be ordered to find out for sure.

However, with the correct testing for each STI there should be a correct result that is certain. Chlamydia and Gonorrhea should be tested from a swab cultures, viral STI's should be detected by blood.