Some will, some won't
Flight schools in first world countries such as the USA and Europe will start to take strain as the demand for pilots dries up and less people are able to afford to learn to fly.
Additionally, a lot of flight schools in the USA have relied on the rapidly expending 3rd world countries to supply them with students, but the current financial crisis has resulted in currencies from 3rd world countries or emerging market economies weakening considerably against the 1st world currencies, making flight training in the USA and Europe unaffordable.
However, flight training in emerging market economies is now very affordable due to the weakened currency and, as an example, a Multi Engine CPL in South Africa, with accommodation, can be completed for less than $30,000 USD. The license is highly respected world wide and ICAO recognised.
What is a worry is that some schools, as they become desperate for students and revenue, will resort to underhanded tactics to entice students to join them and will then not deliver the goods.
The best advice is to never pay a large deposit in advance of commencing training but rather opt to pay in small amounts as your training progresses. If they ask for large deposits in advance, smell a rat!!