The BBC found the authority has taken more than a year to answer five requests, which should have been dealt with in 20 working days.
Alex Runswick, from Unlock Democracy, said long delays gave the perception the authority had "something to hide".
The council said it was working to improve the situation.
Bristol City Council added its responses were "not at acceptable levels".
Ms Runswick said releasing information was "fundamental" and helped "hold government, at all levels, to account".
She said the situation was "bad for our democracy".
"Freedom of Information should be a positive way for councils to engage with people they serve, not something they try to hide from."
During 2014 Bristol City Council responded to fewer than 50% of its 1,955 applications within 20 working days.
In 2013, the figure was 60% of its 1,501 requests.
The Information Commissioner's Office (ICO) states an authority will be scrutinised if "it appears that less than 85% of requests are receiving a response within the appropriate timescales [20 working days]" or when numbers "have gone significantly over this time limit".
An ICO spokesman said the delays "clearly raise questions" about the council's performance and would examine the figures before deciding whether the authority needed to be placed on formal monitoring.
Governmental departments are required to publish figures relating to their FOI performance but there is no such requirement for councils to publish the information.
At present, the ICO is monitoring the "timeliness" of Cumbria County Council, Nottingham City Council, Salford City Council, and the Department of Finance and Personnel (Northern Ireland).
Long delays in answering Freedom of Information requests by Bristol City Council "are bad for democracy", a campaign group has said.