One resident of the Dewhirst Buildings in Kirkgate claimed things have got so bad, rats jumped at his leg when he approached the bin store, while the building's corridor and fire exit are now being clogged with trash.
Another resident says the growing garbage pile "absolutely stinks", and claimed the building "is becoming like a landfill".
"There are lots of rats - they are genuinely everwhere. I have had one jump up at my leg and I had to kick it away. The bins are right next to the door, so they get startled because you are so near them. Even when you walk past the bins, you can hear them rustling.
"When I tried to get in contact with the council, you are often waiting on the phone for two hours, and I work full time. I would like to know why it's happening - I know the council has a shortage of bin drivers, but its become hell on earth."
Siobhan, a teacher who has lived in the in the Dewhirst Buildings since August last year, said: "The collections used to be regular, but this has been going on for about a month. It absolutely stinks, and I don't know where we are supposed to put the rubbish - it's becoming like a landfill.
"It's now in the corridors - this is not good at all. One of my neighbours has started taking his bags of rubbish to work so he can dispose of them there."
"I just want it all removing and for some consistency back - it's got to the point where it's putting me off putting my bins out."
Leeds City Council have been contacted about the situation, but are yet to respond.