The voicemail box stores up to a set number of messages, whether new or saved, so you'd want to delete some (prompt 7) from your mailbox to stop it from being full to make room for new ones to be left.
If you have some voicemails you want to keep like I do just save them to sounds and then delete them. They will remain in your sounds folder but deleted from voicemail thus freeing your space. I have Premium Voicemail which makes the whole process easier since I don't have to listen to the voicemails. I have my 11 year olds first Merry Christmas Grandma! Also sounds transfer from phone to phone so you never lose them!