There is no rule. Every place has its own unique gentilic for the name of the people from there, which is also known as a demonym. You simply have to look each one up individually, each and every time.
- Antigua and Barbados: Antiguan or Barbudan
- Bosnia and Herzegovina: Bosnian or Herzegovinian or Bosnia(n)-Herzegovinian [thanks JBJ]
- Saint Kitts and Nevis: Kittitian or Nevisian
- Saint Vincent and the Grenadines: Vincentian or Vincy
- São Tomé and Principé: São Toméan or Santomean
- Trinidad and Tobago: Trinidadian or Tobagonian or Trinbagonian
- The Turks and Caicos Islands: Turks and Caicos Islander
In a few cases, people have tried to combine them, but that doesn’t look too nice: consider Trinbagonian or Bosnian-Herzegovinian.
Sometimes only one part of the double-barrelled part is used, such as with Santomean, which skips the Príncipe portion altogether.
But if they are actually two places, each place gets its own gentilic. So for example if you are from Antigua and Barbados, you are still either an Antiguan or else a Barbudan, not an *Antigua-Barbudan.
The real question is not the ands but the ors. In other words, what do you call someone from Truth or Consequences, New Mexico? Best to just say that somebody is from T or C and leave it at that.