If the Soldier has completed basic training, AIT, and is assigned to a regular unit then Yes they can be flagged for failing the APFT or HT/WT standards.
The specifics references can be found in AR 350-1, AR 600-9, and AR 600-8-2
G-9m(10)(a) Soldiers in basic combat training must attain 50 points on each event or a score as determined by HQDA, DCS, G–3/5/7, in coordination with TRADOC. (This exception does not apply to AIT, OSUT).
G-9m(6) Soldiers who fail a record APFT for the first time or fail to take a record APFT within the required period will be flagged in accordance with AR 600–8–2. In the event of a record test failure, commanders may allow Soldiers to retake the test as soon as the Soldier and the commander feel the Soldier is ready. Soldiers without a medical profile will be retested no later than 90 days following the initial APFT failure. Reserve component Soldiers not on active duty and without a medical profile will be tested no later than 180 days following the initial APFT failure.
3-2a. Soldiers are required to meet the prescribed body fat standard, as indicated in appendix B. Soldiers will be screened every 6 months, at a minimum, to ensure compliance with this regulation.