Out of 15 cars that completed flying laps in the first session, Button and Magnussen were the slowest and had precious little time on track for technicians to gather data.
They were to gain little cheer from the second session, with former world champion Button improving only to 13th quickest.
Magnussen, standing in for the injured Fernando Alonso, was last-placed at 16th and forced to walk off the track after dragging his car into the gravel.
Strugglers Manor Marussia, recently reborn after going into administration, may have saved McLaren further blushes by being unable to put either of their cars onto the track.
McLaren may not have won a race since 2012 but the prospect of both drivers failing to pass the first round of qualifying on Saturday has become frighteningly real.
While the team's much-hyped partnership with Honda has failed to deliver a competitive car in time for the opening race of the season, management have also been under fire from media speculation about Alonso's health.
The two-time world champion had a heavy crash at testing last month and suffered temporary memory loss.
The team blamed high winds for the collision rather than mechanical failure but the incident is still being probed by the FIA.
Racing director Eric Bouillier forecast weeks ago that McLaren might need until the Europe swing in May to be competitive and on Friday could only speak of what might be.
"Clearly, our practise times today indicate that we have work to do in order to close the gap to our rivals. It was a difficult day, but we expected that," he told reporters.
"We are still in the process, as you can see today, of... understanding the car. We still have a lot of parameters to work on and discover.
"I don't know how long it's going to take."
Magnussen, who was runner-up for McLaren last year at Albert Park in a stunning F1 debut, blamed himself for his skid off turn six and minor collision with a barrier, adding his car had "balance" if not speed.
"The positive is that the balance and feeling of the car feels good," he said. "Clearly, there's a lot more to come but it's definitely a good baseline."
(Editing by John O'Brien)
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