It took us
7 hours to drive from San Antonio to BBNP when we headed there in October 2001 for a week of hiking and camping (rented a pop-up camper).
NOTE: that's 7 hours just to get from northern San Antonio to the park's entrance station. The nearest vistor center is in Persimmon Gap,
another 26 miles (at 45 MPH) down the road. During our visit, the Persimmon Gap Visitor Center was closed; when it's open, you have the opportunity to stop there and check in, rather than travelling all the way to Panther Junction (located about halfway into the park).
I wouldn't do it for a weekend because to explore the park you'll find yourself doing a lot of driving even once you get inside the park's borders... doing that ALL weekend long would be tiring and take away from time spent, ostensibly, doing what you've come to enjoy -- hike, take photographs, observe the wealth of desert wildlife and plant life.
Holiday weekends should be avoided, as well as Spring Break, if one is visiting with the intent to enjoy a quiet, park-like atmosphere. Still, it's a BIG park with less visitation than most national parks, so if one stayed to lesser-travelled trails I can't imagine it'd be akin to a Yellowstone in summer experience (think Disneyland).
There were a fair amount of RV campers in the Rio Grande Village Campground during our October stay, but it was by no means "busy". Only on the last day of our trip did we encounter anything approaching a "crowd" and that was only because the local kids from across the border were coming into the park to perform an annual play/festival. (I'm not sure they do that anymore, what with the stricter border enforcement; they shut down the border crossing within the park -- used to be a popular way for Big Bend NP visitors to grab "authentic Mexican food" from Boquillas, Mexico -- a town that's pretty much cut off from the rest of the Mexican nation, due to the Sierra del Carmen mountain range.