Yes, they can, IF:
1) It’s provided in the lease contract, and
2) It is given to the tenant a 30 days-notice of this increase.
This way, for instance, if it is provided in the lease contract a €500 rent per month on September 2023, during the last days of July of 2024 landlords would have to send a reliable communication to their tenants informing them that from September 2024 the rent will be €500, plus the CPI increase, in this case, 2.1% and for that reason, the monthly rent from September 2024 onwards would amount to €510,50.
Landlords may bear in mind that the Government has limited this increase up to 3% for the contracts to be signed until the December 31th 2024.
Can the rent raise cumulatively, if the previous years this increase has not been applied?
The CPI % could be raised cumulatively, but in no case, you could claim the difference of the increases for already-expired month on the rents for which you have not requested it in that moment.
For example, if the landlord did not update the rent to be paid by its tenants in 2023, the landlord could apply an accumulated increase in 2024. In other words, the landlord could increase the rent by 5%, broken down into: 2% CPI Increase corresponding to the limit for 2023, and the 3% CPI increase for 2024.
However, claiming the difference of what you would have received in 2023 would not be possible, if the rent was not updated in time.