Yes it is possible to get chicken pox twice. My son got them just before his first birthday, then again when he was about six.
And guess what! He got married nearly 3 years ago and did not dare tell his bride that he felt ill as he did not want to worry her, so the day of the wedding he went to the church feeling awful.
After the ceremony he told his new wife and they went on honeymoon only to return the next day as he had shingles which of course is the same virus.