Veterans Education and Transfer Extension Act or the VET Extension Act This bill makes changes to Post-9/11 educational assistance for veterans or active duty military service members. Specifically, the bill increases the number of months of educational assistance available under the Post-9/11 GI Bill by (1) 15 months, or (2) the amount of months needed to complete remedial and deficiency courses that may be needed prior to beginning a program of education. Under the bill, individuals who do not currently have a dependent, but wish to elect to transfer educational assistance to a dependent, may do so at such a time when they do have a dependent. Additionally, the Department of Defense may not prescribe any additional limitation on the time period in which a member of the Armed Forces may make an election.