First thing: will Tioga Road be open? Maybe. If so - will the snow levels be such that you are comfortable hiking through some of it? They plow the road, so even if the road is open, there will still be snow on trail, and high water levels due to snowmelt.
Second thing: will the bridges at Pate and at Return/Register creek be in good repair to make the crossings? Would be dangerous without them.
third thing: Will White Wolf the resort be open? aka, where are you going to be able to leave a car, or are you willing to walk all the way out to Tioga Rd if WW is closed, and chance being able to flag down a shuttle/bus?
Have a look at the opening date chart:
http://www.nps.gov/yose/planyourvisit/seasonal.htm
White Wolf is typically late June openings. Tioga Rd has been known to open late in June.