Both the International Federation of Gymnasts and the International Tennis Federation said no rules had been broken.
The hack on the WADA is thought to have been carried out in revenge for Russia being banned from the Olympics in Rio.
The Kremlin was furious at the treatment of its athletes in Rio with some accusing Olympic chiefs of operating 'double standards' in banning Russian athletes while allowing US athletes previously found guilty of doping to compete.
WADA confirmed Russian hackers broke into its database and posted some athletes' confidential medical data online.
The hackers revealed records of 'Therapeutic Use Exemptions' which allow athletes to use substances that are banned unless there is a verified medical need.
WADA previously warned of cyber-attacks after investigators it appointed published investigations into Russian state-sponsored doping.
Director general Olivier Niggli said: "These criminal acts are greatly compromising the effort by the global anti-doping community to re-establish trust in Russia."
"WADA condemns these ongoing cyber-attacks that are being carried out in an attempt to undermine WADA and the global anti-doping system.
"WADA has been informed by law enforcement authorities that these attacks are originating out of Russia."
Fancy Bears claims it had more information showing how 'tainted' Olympics medals have been won.
It said: "Wait for sensational proof of famous athletes taking doping substances any time soon."
The International Olympic Committee said it 'strongly condemns such methods which clearly aim at tarnishing the reputation of clean athletes.'
An IOC spokesman said: "The IOC can confirm however that the athletes mentioned did not violate any anti-doping rules during the Olympic Games Rio 2016," the Olympic body said.
"Fancy Bears," is apparently a tongue-in-cheek reference to a collection of hackers which many security researchers have long associated with Russia.
In a statement posted to its website early Tuesday, the group proclaimed its allegiance to Anonymous, the loose-knit movement of online mischief-makers, and said it hacked WADA to show the world "how Olympic medals are won."
Internet records suggest Fancy Bears' data dump has been in the works for at least two weeks; their website was registered on September 1 and their Twitter account was created on September 6.
It is the latest significant leak by Russian-based hackers suspected of links to the Moscow's intelligence services after a group broke into the US Democratic Party's National Committee computer networks and gaining access to data relating to Donald Trump.
SUBSTANCES WILLIAMS ARE ACCUSED OF USING
Serena: Oxycodone (2010), hydromorphone (2014), prednisone and methylprednisolone (2015).
Venus: Prednisone, prednisolone, triamcinolone and formoterol.
WHAT ARE THERAPEUTIC USE EXEMPTIONS?
Therapeutic use exemptions are granted to athletes with health conditions that require them to take normally banned drugs.
These allow them to take substances found on the WDA's prohibited list without fear of punishment.
The exemptions, which often attract controversy, have previously been granted to cyclists who say they have asthma.
Another sport that has seen several exemptions granted is MMA (mixed martial arts).
Some fighters have been allowed to take steroids because of low testosterone.
