Volunteers for Rape Crisis Scotland handled a record number of phone calls last year and the charity warned funding shortages mean it is struggling to cope with demand.
Some callers are greeted by an engaged tone or an answering machine message because the lines are jammed and there are waiting lists for face-to-face counselling, according to the charity.
The Edinburgh office, which has more clients than any other, already has a nine-month wait for counselling sessions and will lose half its funding when a National Lottery grant runs out in March.
Sandy Brindley, national co-ordinator of Rape Crisis Scotland, said: "We are really worried about what that will mean for people trying to access services in the area."
Across 13 Scottish centres, the charity handled 12,622 calls last year from 1697 victims. This is up from a high of 12,418 in 2011.
Nine out of 10 calls last year were from women, three-quarters of the callers had not reported the crime to the police and more than 85% were attacked by someone they knew.
Ms Brindley said: "People still think rape is about a stranger jumping from behind a bench and it is not that serious if it is carried out by someone you know. This new report shows it [the attacker] is primarily someone people knew and it also shows how devastating sexual violence can be whether it is carried out by a stranger, or a partner or a family member."
One-fifth of the victims of rape and sexual abuse had tried to kill themselves, according to Rape Crisis Scotland. Almost half had suffered flashbacks and depression. Nearly 40% had dreams or nightmares triggered by the experience, 27% described eating problems and 26% self-harm.
Rape Crisis Scotland is asking the Scottish Government to ensure there is more funding to support the work it does. At present, the Government gives each centre £50,000 a year.
A Scottish Government spokesman said: "Rape is a horrific crime and the Scottish Government will continue to work with partners to improve the support available for all victims."