Is it possible to have a false negative syphilis test?

Yes, early infection. It takes a few weeks for the blood tests to become positive, so false neg is common soon after infection. Once infction is established and a few weeks have passed, false negatives are rare. And even in those rare circumstances, the infection usually is not active or likely to be damaging. If outwardly healthy, a neg blood test is virtually 100% assurance against active or transmissible syphilis.
Yes. Testing for syphilis can sometimes be a complicated process which requires expert interpretation. Unfortunately there is no test in medicine which is 100% accurate 100% of the time. A false negative syphilis test may occur within the first 2 weeks after the common syphilis ulcer appears. Patients with immune disorder such as HIV and aids may also test falsely negative for syphilis.