“At the same time [Felix] was hoping that Paul would give him money. For that reason, he sent for him even more frequently and conversed with him. But when two years had elapsed, […] he left Paul in custody.”
~Acts 24:26,27

How can a Christian distinguish between giving a bribe and tipping an official to ensure a service is rendered?
The Bible clearly condemns bribing. (Ps. 15:1,5)
But what could be considered a bribe in one country, could be considered a customary tip in another.
I remember a traffic officer in Mexico who would not release us until my aunt (not one of Jehovah’s Witnesses), implied she would give him a tip.
On other occasions in which my husband and I were pulled over, we accepted citations for minor traffic infractions instead of handing over any cash.
But it is true that many government officials, especially in developing nations, do not make enough money to live on, so whether or not a Christian decides to tip one is a matter of personal conscience. (Mark 12:17; 1 Cor. 10:31-33)
It would be blatantly wrong to give something with the intent of evading justice or seeking preferential treatment over others(Deut. 16:19; Matt. 7:12)
Despite his reputation for corruption, Felix as governor did have a legal right to hold Paul indefinitely without handing him a verdict. (Watchtower. 2001, December 15. “I Appeal to Caesar!”)
If Paul had caved in to bribing him, he would have been breaking Roman law.
As Christians, we find comfort in knowing that Jehovah will bring ultimate justice and he cannot be bought. (Deut. 10:17)