Bester appeared on Monday in Hobart Magistrates Court where he was expected to face a two-day hearing for allegedly harassing Ms Tame on social media.
The 72-year-old is accused of making public posts in relation to and directed at Ms Tame in April, July and August of 2022.
Bester pleaded not guilty in March 2023 to three counts of using a carriage service to menace, harass or cause offence.
Former Australian of the Year Grace Tame was not in court on Monday but has attended previously. (Ethan James/AAP PHOTOS)
He was jailed in 2011 for sexually abusing Ms Tame when she was a 15-year-old student at the school where he taught and for possessing child exploitation material.
Bester's lawyer Todd Kovacic told the court his client had vascular dementia and his fitness to face a hearing was an issue.
Mr Kovacic indicated Bester's fitness would "almost certainly be conceded" based on medical assessments from the defence and prosecution.
Mr Kovacic was granted an adjournment by Magistrate Andrew McKee so the question of "where to from here?" could be answered.
The matter is expected to next come before the court on August 28.
The Director of Public Prosecutions had requested more information about Bester's medical condition, Mr Kovacic said.
The court has previously been told Bester, who was charged in October 2022, had suffered a "medical episode".
There have also been delays and adjournments in the case to obtain medical reports.
Ms Tame, who was not present in court on Monday but has attended on other occasions, was the 2021 Australian of the Year and has advocated for abuse survivors.
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