NEW YORK—The few years when the late ’60s staggered into the early ’70s were strange days indeed. (And contrary to the old saying, I was there and I can remember.) The exhibition “High Times, Hard Times: New York Painting 1967-1975” offers a fascinating and long-overdue reminder that those days were especially strange for abstract painters.

Looking particularly outrageous in the ornate neo-classical spaces of New York’s National Academy Museum, where the traveling show is on view through April 22, the paintings in the exhibition are yearning to be something else.

Joe Overstreet’s Purple Flight, from the “Flight Pattern” series (1971), is tethered by eight separate ropes and twists in the air above a staircase like a purplish sprayed blanket; Lynda Benglis’ Blatt (1969), in poured pigment and latex, spreads unpleasantly across the floor like some giant kid’s Play-Doh disaster; and Alan Shields’ not-insignificantly titled Put a Name on It Please (1972), made of cotton belting and long strings of love beads, hangs rather limply across the room like a soccer goal net.

These artists questioned just about all of the assumptions of painting, and as often as not jettisoned them. Rectilinearity, rigidity and flatness were among the most obvious characteristics that some of them shed, plus the presumption that a painting should hang on a wall.

The ways in which paint was applied also became something of a free for all. Spraying, throwing, dragging, pouring, stenciling and dyeing were clear favorites. Apparently Lawrence Stafford went so far as to “wield a spray gun to paint vertical bands as the canvas spun before him on a rapidly rotating mechanical drum,” according to the exhibition organizers.

Sadly though, a lot of the pieces that survive from that era are unable to transcend the rather simple questions they ask. Can you paint on the edges of a picture rather than the front? Sure. Can you bind sheets of canvas like a book? Of course. Can you braid lengths of fabric together so that you finish up with something very like a rug? Certainly.

Perhaps at the time it felt like anything was possible, but the sense that you get from this work now is that the reverse was true. Here was a generation for whom painting had run out of possibilities.

In some of the more interesting work here, painting literally does turn into something else: sculpture (in the case of Howardena Pindell’s Untitled [1968-70]), or video (as in Roy Colmer’s Video Feedback, Cincinnati [1972]), or installation (Dorothea Rockburne’s Intersection [1971]), or performance art (Carolee Schneemann’s Body Collage [1967]).

Perhaps it’s not altogether surprising. So far as the histories of that period are concerned, those were the territories in which the most compelling art was being made.

Still, this show provides an important corrective to some of the histories. There was an experimental painting culture in New York City during those years, even if the majority of the work that it produced could show no clear way forward.

When painting once again demands a place in the art mainstream some five or 10 years after this survey comes to a close, it will be on the basis of characteristics, such as representation, expressionism, popularism and romanticism. But those qualities were mostly irrelevant—if not incomprehensible—to the artists of those “High Times, Hard Times.”