During the TNG series, several planets are shown in various stages of the application process for membership with the UFP but we don't see the final outcome of those applications, but it doesn't look good.
The Anticans applied for membership to the United Federation of Planets. The Enterprise-D transported their delegates to the planet Parliament to help resolve a long-standing dispute with their sister planet, Selay, as part of the admission process. (TNG: 1x07, "Lonely Among Us")
The Angosians applied but were apparently turned down. Picard tells the Angosian government that they must make a decision regarding the fate of their veterans and that he has all the information he needs for his report, and, if they wish to reapply for Federation membership, their application will be given all due consideration... (TNG: 3x11, "The Hunted")
Looks like the Angosians got shut down and the Anticans were a real longshot for acceptance. Have there been others who have been denied membership?