Just as in their seven previous campaigns in the competition, they have failed to get off to a winning start.
The Hoops had been making progress, ending a run of five match day No.1 defeats by earning goalless draws against Aalborg in 2008 then Benfica 12 months ago.
But the 2-0 defeat to AC Milan in the San Siro - Cristian Zapata breaking the deadlock after 82 minutes, firing in with the help of deflection from Emilio Izaguirre's right foot, then Sulley Muntari scoring into an empty net three minutes later after Fraser Forster had saved Mario Balotelli's free kick, awarded for an off-the-ball clash with Scott Brown - means it is back to playing catch up from the off.
And with their next match against Champions League second favourites, Barcelona, that's precisely the situation Neil Lennon wanted to avoid.
More frustrating for the Hoops boss is the fact his side did not deserve to leave all three points in the San Siro, so well did they play.
But as they proved impressively last year, there remains a very long way to go, with many twists and turns along the way.
And no-one in Hoops needs any reminding the mighty Barca were one of the giants to fall as Forster and Co. added another nine points to the one they collected from their opener against Benfica as they upset the odds to qualify for the last 16.
To emulate that, from a much stronger group and minus key cogs in the Green Machine of last season, Victor Wanyama, Gary Hooper and Kelvin Wilson, is a massive ask.
But Lennon and his players like nothing better than jousting with giants, and Milan, Barca and Ajax do not frighten them, as last night's battling display confirmed.
The Parkhead boss is still discovering the strength of this squad after the summer's comings and goings.
Despite negotiating six qualification and play-off games, only when the competition proper kicks off can you really gauge what you have got, how it measures against what you had and, even more importantly, against the teams you now face.
Tellingly, of the six recent recruits, only Virgil van Dijk got the nod to start against Milan, which was seen as a vote of confidence in the men who played their part in last season's run.
Anthony Stokes - making his first start in a Champions League match - and Charlie Mulgrew - playing at the base of a midfield diamond - were handed the challenge of filling the void left by Hooper and Wanyama, with newest Bhoys, Teemu Pukki and Nir Biton among the subs.
With Adam Matthews finding space and good possession on the right of midfield, and Izaguirre joining Georgios Samaras raiding down the left, the plan to get crosses into the Milan box kicked into life midway through the first half.
Having sat tight as Milan dominated the early stages, they worked their way into the game and Stokes and Mulgrew, from an indirect free-kick, had good chances to score in the first half, only to see their efforts blocked.
Milan looked more comfortable attacking than defending, with Balotelli and Alessandro Matri proving their multi- million-pound worth as they united in search of a goal to settle the very apparent home nerves.
By half-time, Forster had been called upon to save twice from Balotelli, once from a Matri header, and to dive at the feet of Valter Birsa. But it would have been tough on Celtic had they gone in trailing.
The fact they had taken the game to Milan after a cautious start and were enjoying long periods of possession was to their credit, and clearly upset the Rossoneri side which, though shorn of several big names, was still studded with stars and underpinned by quality.
In the face of this, the only negative which could be levelled against the Hoops was the quality of their final ball and execution of their set pieces.
In the plus column, they looked comfortable in possession and organised when not, which perhaps explains how relaxed Lennon looked in the technical area.
His heart must have skipped a beat when Muntari found himself unmarked six yards from goal, but the midfielder sent his header well over the bar.
Lennon was much happier as he watched his players unite to ask serious questions of Milan.
The home fans feared the worst when twice Samaras had good chances to score, only to see his shots blocked for corners.
Even Brown got in on the act, bursting a gut to get on the end of a Commons pass in a breakaway, forcing Nigel De Jong to make a last-gasp tackle.
The introduction of Derk Boerrigter and Pukki for Matthews and Commons indicated Lennon's eagerness to go for all three points, and Stokes almost delivered, hitting the top of the bar with Abbiati beaten after Pukki had drawn a free-kick 25 yards out.
But the sucker punch awaited, courtesy of Zapata, Izzy's toe and Muntari.
At least Ajax's hammering from Barca means the Hoops are not bottom of the group. But that's little consolation for Lennon and his players.