If you allow me, it works fine also for multi-attachments, the 1st above answer of NINCOMPOOP, with just a little modification like follows:
DataSource source,source2,source3,source4, ...;
source = new FileDataSource(myfile);
messageBodyPart.setDataHandler(new DataHandler(source));
messageBodyPart.setFileName(myfile);
multipart.addBodyPart(messageBodyPart);
source2 = new FileDataSource(myfile2);
messageBodyPart.setDataHandler(new DataHandler(source2));
messageBodyPart.setFileName(myfile2);
multipart.addBodyPart(messageBodyPart);
source3 = new FileDataSource(myfile3);
messageBodyPart.setDataHandler(new DataHandler(source3));
messageBodyPart.setFileName(myfile3);
multipart.addBodyPart(messageBodyPart);
source4 = new FileDataSource(myfile4);
messageBodyPart.setDataHandler(new DataHandler(source4));
messageBodyPart.setFileName(myfile4);
multipart.addBodyPart(messageBodyPart);
...
message.setContent(multipart);