The Ivorians were held 0-0 by Togo — 21 places below them in the African rankings — after a duel that produced little excitement for a small crowd in the northern town of Oyem.
Morocco, also considered potential champions under twice Africa Cup-winning coach Herve Renard, paid for a goalkeeping blunder as they fell 1-0 to the Democratic Republic of Congo.
DR Congo, whose last of two Cup of Nations triumphs came 43 years ago, surprisingly top Group C after the first round with three points.
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Ivory Coast and Togo have one point each and Morocco, whose lone Africa Cup success was in 1976, are pointless at the bottom.
Group winners and runners-up after a three-round mini league advance to the quarter-finals, leaving Renard with just two games to perform a salvage operation.
The Oyem results followed a string of other unexpected scorelines in the biennial showcase of African football, which kicked off on Saturday.
Guinea-Bissau and Burkina Faso came from behind to gain unexpected draws with Gabon and Cameroon respectively in Group A on the opening day.
Zimbabwe, considered a lightweight among the 16 teams at the tournament, led star-studded Algeria for much of a Group B thriller before pulling off a surprise 2-2 draw.
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DR Congo are only the second side to collect maximum points so far, following Senegal, who overcame wasteful Tunisia 2-0 in Group B on Sunday.
"It is not the start that we would have liked; we could not raise our level of play," admitted Ivory Coast coach Michel Dussuyer, who succeeded French Ligue 1-bound Renard after the 2015 triumph.
"There are still two matches left. We expect more from ourselves. We did not manage to put intensity into our game. We played at 1700 local time and it was very hot."
Togo coach Claude le Roy was satisfied with a point as he began his ninth Cup of Nations having reached the knockout phase in seven previous campaigns.
"I feel we had a good match. Hats off to my players. We qualified [for this Cup of Nations] at the last second in a country where there has not been a championship for the past three years," said Le Roy.
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"I feared this Ivory Coast team. On some actions, we were near scoring, which would have been marvellous."
Midfielder Junior Kabananga pounced on a goalkeeping error to give DR Congo a 1-0 win over Morocco.
It came two days after the central Africans refused to train over a bonuses row and a red card and an injury meant they were briefly reduced to nine men in the closing stages.
Kazakhstan-based Kabananga profited from a mistake by Munir Mohamedi in the Morocco goal to score from close range 10 minutes into the second half.
The Congolese were reduced to 10 men with eight minutes left when a wild challenge from substitute Joyce Lomalisa earned a second yellow card.
They were then temporarily left with nine men on the field when injured captain Gabriel Zakuani had to receive treatment.
"What we had to do was be patient," said DR Congo coach Florent Ibenge. "We knew that it could pay off if we were very strong behind."
Renard said: "I think we got off to a good start. We caused them a lot of problems. We had chances, but were not efficient when it counted. In a competition like the Cup of Nations you must not get discouraged under any circumstances. There will be a lot of developments."
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