Citizens Advice Scotland said people are waiting an average of six months between claiming the new Personal Independence Payment (PIP) and receiving the payment.
This can have a knock-on effect on other benefit claims, it said.
PIP has been introduced in Scotland over the past year.
It is part of wider welfare reforms by the UK government, and has replaced the Disabled Living Allowance (DLA) as the main benefit that helps disabled people meet the costs of their basic daily living.
Citizens Advice compiled a report on the implementation of PIP using evidence gathered from its Citizens Advice Bureau staff.
It found that people have experienced average delays of six months, but some have reported delays of 13, 14 or 15 months.
Seven out of ten advisers surveyed said they are dissatisfied with the PIP claim process, particularly as a result of delays.
They also reported problems with the assessment process, with almost a quarter of advisers stating that some decisions are inaccurate and need to be appealed.
More than half of the advisers said the delays cause the benefit claimant “severe hardship”.
PIP is also a passport to other benefits, therefore the delays mean that some claimants are also missing out on the financial support of other benefits.
Citizen Advice Scotland’s head of policy Susan McPhee said the basic cost of living is generally higher for sick and disabled people than for the average citizen.
This is because of the additional costs of special food, medicines or equipment they might need, extra heating and lighting costs for those who need to stay at home longer, or transport costs for those who are less mobile.
She added: “These extra costs are not luxuries. They are essential to leading a basic life of dignity and any civilised society should make it a priority to see that people who need this help get it without fuss.
“The evidence we are publishing here shows that, under the new Personal Independence Payment, too many disabled people are not getting that support and many are falling into poverty as a result.
“While some claimants are reporting that the system has worked well for them, we are still finding too many delays and too many inaccurate assessments.”
Ms McPhee said it was “deeply worrying that so many problems have already emerged with the new system at such an early stage.”
She called on the government to urgently address these problems “as they are causing great detriment to some very vulnerable citizens and their families.”
In June, the Public Accounts Committee at Westminster warned that the PIP scheme had been “rushed” through, with a “shocking” impact on claimants.
But Mark Harper, UK Minister of State for Disabled People, said: “Unlike the old system, Personal Independence Payment includes a face-to-face assessment and regular reviews to ensure support goes to those who need it most.
“The latest figures show just that, with nearly 23% of people getting the highest level of support, compared to 16% under DLA.
“Between May and July we have doubled the number of claims processed and we are working hard to continue to make further improvements.
“By the end of the year we expect that no-one will be waiting for an assessment for longer than 16 weeks.”
But SNP MSP Jamie Hepburn said the report was “damning evidence of the devastating consequences Westminster’s welfare cuts are having on thousands of sick and disabled people in Scotland.”
He added: “The Westminster establishment have shown time and time again that they can’t be trusted on welfare – their race to the right to court votes in the south east of England is punishing the very people who need support the most – and the Tory conference this week has only reaffirmed their commitment to a shameless and sustained attack on the poor.”