i have the same problem
i have a s4 mini also with sprint i wonder if it might be a sprint issue as opposed to a twitter one
Thought its a sprint issue, as i asked a few friends that had the same problem, & & they were on sprint, but later a few people with verizon told Mr that they have the same problem, so it's probably a twitter issue.
(I tried sending the messages to a google voice number, & it should get forwarded to my cell phone, but it didnt help, so u see that also GV has this prob, not only sprint)