Do I have to remove all caulk before reapplication?
If only a small section of the caulking has opened it may be worth re-applying fresh caulk to them with the hopes that it won't happen anyplace else. Since the repair is small cut out the compromised caulk and clean the surrounding areas with denatured alcohol. Apply fresh caulk forming it into place to match existing caulk. FYI: It is recommended, but not required, for longevity and resiliency to use a 100% silicone caulking. Pure silicone won't dry and become brittle in time and are able to flex if any of the two materials (wall and/or tub) move or shift. If you are adamant in using latex or a siliconized  caulking use one which is advertised for exterior use (the quality brands will be touted as having a 30-40 year guarantee, usually for doors and windows). Also, if you can find it, look for a mildew-proof or resistant-type caulk. I've re-caulked jobs several times because mold had established itself on and in the caulk.