Check out the website http://www.nps.gov/yose/planyourvisit/wildcond.htm
As of March 3, 2016, the website said this about General Conditions under the heading Wilderness Conditions
Be prepared for winter conditions throughout the park. The current
snowpack is about 85 - 100% of average for this time of year,
depending on location, a significant contrast to the last 4 winters of
50% or below snowpack. With spring around the corner, lower
elevations, such as Yosemite Valley and Hetch Hetchy, are mostly snow
free. However, there is still significant snow at higher elevations.
The snow line is highly variable due to aspect, vegetation and
exposure to the sun, but expect significant snow above 6,000 ft. In
shaded north facing areas, expect snow at lower elevations. There are
multiple feet of snow at elevations of 7,000 ft and above
Based on long experience with Yosemite, a snowpack of 85 to 100 percent of average means that Tioga Pass Road will not be open before May 15, and Glacier Point Road probably not before May 1. Even that is optimistic. The website used to have a list of the dates these two roads opened, but I have not located it in my brief search for you.
You can get to the snow-gate at Crane Flat (which is roughly 6,500 feet) on Tioga Pass Road; there are trails off Tioga Pass Road before the snow-gate which are fun to x-country ski or snow-shoe.
The section on Trails and Wilderness says:
The Mist Trail from its junction with the John Muir Trail above Vernal
Fall footbridge to the top of Vernal Fall is open; the John Muir Trail
from Clark Point to near the top of Nevada Fall is closed for the
winter. It is possible to hike to the top of Vernal and Nevada Falls
using still-open sections of trail, however, conditions may be snowy
I've hiked from the Valley to the Rim on the Snow Creek Trail in late April, but late March..... Much depends on your level of ability. There are people who would think nothing of getting to the High Country from the Valley in January.
You can't count on the weather. How long would you be there?
My bottom line is that, in late March, you may be confined to the Valley plus maybe the first 1,500 feet above the Valley plus the open part of Tioga Pass Road. And there may be weather for a day or three. On the other hand, the Valley will be spectacular and uncrowded and the waterfalls should be awesome. There is nothing like Yosemite. Note: Many, maybe most, visitors never go beyond the Valley.