I've seen it myself. On arrival at their VFR school on the other side of the world I waited a month to be allocated an instructor. A whole month of just sat in the accommodation. Waiting. No emails. calls, apologies or anything. Instructors were resigning by the handful, and still dozens of students were arriving every couple of weeks. At a point my instructor had 13 students. I flew once per week if I was lucky. But that ended up once every 10 days or even two weeks, when the tagged students had been prioritised. We were quoted 5 months for this stage, but it took me just over 7 months. I passed every check ride first time and did the course in minimum hours, so I didn't cause any delays myself. Accommodation was part of the fee, but it was still two more months of living costs to pay for. As Tecnamflyer says, the finance department were pestering us for payment of our fee instalments after a month of us being sat around, waiting to be allocated an instructor so we could actually start our training. It was abysmal.
On arrival back in the UK for the IR stage, it was slightly better. But then a load of instructors resigned, and the students kept pouring in. Instructors were used to 2-3 students each, but this soon became over 6. When I had my IRT, my instructor had 9 students. Flying once per week was the norm. Eventually the school started giving students leave after retiring from the VFR stage abroad, and starting their IR. To restrict the huge flow of students into the severely understaffed IR training department. It started as a month, and when I finished was more like 4 months. As far as I know, it could be more than that now.
If you're self-sponsored, then you're the bottom of the pile. The sponsored schemes and tagged students always have the upper hand, especially when the airlines start getting involved and make complaints. It was common for a tagged student to fly five times per week, whilst a self-sponsored would be once every 10 days.
It was a sorry state of affairs. 90% caused because of a lack of instructors, and 10% poor management. I would seriously consider your actions before choosing to train at a premium integrated school at the moment. The timeframe they quote in the glossy brochures is fantasy. 50 or so more instructors per school would eradicate this problem.