Yes. A pivot shower door can leak, and water has two ways out: past the vertical edges, where the door needs clearance to rotate, or at the bottom pivot, where the door meets the curb. A door whose top and bottom pivots have drifted out of line opens both paths wider, so alignment is the first thing to check.
A pivot door turns on points above and below it rather than on hinges along one edge. The National Glass Association's heavy glass door manual, written for commercial entrance doors but describing the same mechanism, says a swing door is usually attached at pivot points at both top and bottom and rotates on that pivot axis, with the pivot carrying the door's weight. A side-hinged shower door is different: its hinges fasten along one edge to a wall or a fixed panel, so that edge stays put. On a pivot door, both vertical edges move, and each needs clearance so the glass can turn without binding.
Those clearances are the first water path. The same manual sorts pivots into center hung and offset types, and notes that weatherstripping is difficult on center-pivoted glass doors, while offset pivots can allow compressible bulb weatherstripping. The manual is about entrance doors, not showers, but the point carries over: an edge that swings past its neighbor is harder to seal than one that closes against a stop. The National Glass Association's shower guidance is plain about the wider point: a frameless shower "is not an aquarium," and how much one leaks depends mainly on its configuration and the type of enclosure.
The bottom pivot is the second path. On a framed or semi-frameless pivot door, the bottom pivot sits in a bracket or rail fastened to the curb, right where water runs off the glass. If the sealant around that bracket or its fastener holes has let go, water follows the metal out of the shower.
Finding a pivot door leak starts with watching where water lands. A drip at the base of the door's pivot side points to the bottom bracket or its sealant. A trickle down the latch edge or the far edge points to a seal that has gone hard or no longer reaches. Water spread along the whole curb points to the sweep. Seals and sweeps are replaceable wear parts, and diagnosing a shower door leak in general, sweeps and silicone included, is covered separately.
Then check the line. The National Glass Association's commercial door manual says the closer and pivots must be installed plumb to each other and level, and that some swing door hardware, such as pivots, may permit adjustment. A shower door has no closer, but its top and bottom pivots still have to line up. A pivot door that has drifted shows it as a wedge-shaped gap, wider at the top or the bottom of the edge, and a door out of line stops meeting its seals evenly. Whether a particular pivot adjusts, and how, is in the hardware maker's instructions. Frameless hinged doors and semi-frameless channels leak in ways of their own, and each has a page of its own.
Some pivot hardware lets a door swing both in and out, and some limits it to one direction; the hardware maker's specification says which. A door that swings both ways has no stop to close against, so keeping water in depends on its edge seals and on aiming the shower head at the tile rather than at the glass.
Not in the way a sweep is. The pivot is hardware that carries the door's weight and lets it swing, so it is expected to last, while the seals and sealant around it wear out. A worn pivot shows as a door that drops, drags or will not hold its line, and whether parts are still made depends on the hardware maker.
A photo of the bottom pivot, the gaps along both edges and the spot where water lands shows a lot before anyone visits. Text it to Jessica for a fast quote from photos, then book the free in-home measure if new glass or hardware is the answer.