Why can’t people leave messages in my voicemail?

HI Beatlemania1952,

Are they receiving any error messages? Such as 'mailbox full'? Then you just need to erase some of the stored voicemails (or empty the voicemail trash). Launch the Phone app > Voicemail tab at bottom right > Click Edit on top right > Select the message you are ok to remove > tap "Delete".

If there is no error message, then you'll need to contact your carrier. Voicemails are enterly handled on their end. The iPhone just stores the recording.