The consumer group's researchers posed as new customers to ask 13 banks and building societies basic questions about how the Financial Services Compensation Scheme (FSCS) works. Under the scheme, up to £85,000 of someone's savings is protected if their provider goes under.
Researchers told the providers they had £100,000 to deposit. But out of the 156 anonymous calls they made across 13 providers, not one member of staff gave an unprompted warning that £15,000 of this money would not be protected in the event of a collapse.
The 13 providers contacted were Royal Bank of Scotland (RBS), NatWest, Bank of Scotland, HSBC, Lloyds Bank, Nationwide Building Society, Barclays, Santander, Co-operative Bank, First Direct, Halifax, Britannia Building Society and Yorkshire Building Society.
Researchers asked staff specifically about savings protection limits, and those at Yorkshire Building Society and Halifax were the only ones to give the correct answer.
One NatWest employee told a researcher: "I imagine it's like PPI. The bank wouldn't have anything to do with anything like that."
Nearly two years ago, rules came into force which mean banks, building societies and credit unions must display stickers or posters to help raise consumer awareness of compensation levels.
Richard Lloyd, Which? executive director said: "It is inexcusable bank staff can't give customers basic information about the compensation scheme if their bank goes bust.
"In the event of a collapse, this bad advice could cost people many thousands of pounds from their life savings. We hope this is a wake-up call to the banks that they need to improve staff training."
The £85,000 limit applies per banking authorisation rather than per banking brand, so if you have savings with more than one member of the same banking group, your cover may be more limited, although this is not always the case.
For example, Which? highlights that Co-operative Bank, Britannia and Smile share the same authorisation, but RBS and NatWest, which are both part of the RBS banking group, have separate authorisations. For joint accounts, the savings compensation limit doubles to £170,000.
Which? argues the FSCS protection rules would be much easier for consumers to understand if they applied per individual brand.
Its researchers asked bank and building society staff if having savings elsewhere affected levels of protection and found that "overall, the responses were poor".
HSBC came bottom of Which? research, with a score of 51% and Yorkshire Building Society came top with 88%, followed by Britannia Building Society with 82% and Halifax with 81%. Banks said they would offer additional training.
A spokeswoman for the British Bankers' Association (BBA) said: "All employees are made aware of the FSCS and many banks have relevant training, so it is disappointing that more staff did not communicate the benefits of the scheme when asked.
"The industry strives to make sure that detailed information on the FSCS is available via posters in branches, information on bank statements and marketing materials.
"The BBA will work together with the banks to look at the concerns raised in this survey to help make sure customers are given consistent and accurate information on FSCS protection."
A checker can be found at www.fscs.org.uk/protected