At what time frame is there a significant increase of the price of airplaine tickets?

As a whole airfares do not increase, rather available fare classes sell out or become invalid. On average, a economy section can have 4 to 8 fare classes available, with a limited number of spaces available in each class. As the cheaper spaces get sold, then the reservation system shows the next higher fare that is still available.

In years past, a fair percentage of discounted fares were based more on how far in advance you bought the ticket, hence the cheapest tickets you bought 60 days in advance, then another break at 30 days, 21 days etc. But in today's market advance purchase rules have become less important and are not really a factor in most fare classes.

As such there is no longer a rule of thumb on purchase time. Lower fare classes can sell out well in advance or still be available just a few days before departure. Airlines can also add additional spaces to a lower cost fare to boost sales if a flight is not selling well.

One time based factor that seems to apply across many carriers, spaces on flights are usually not available for sale more 330 days in advance. And the most highly discounted fare classes are usually not available for purchase more than 6 months in advance.

Of course for each of these statements, I am sure there is an airline or two who do it differently.

I am afraid the answer is going to be “it depends”.

With low cost airlines at least, the increase is gradual and it's not uncommon to see markedly lower prices for flights at inconvenient times. So a ticket on such a flight might still be available at a given price a week out whereas similar tickets for a more attractive flight on the same day disappeared a few weeks before. In those conditions, it's very difficult to find any discontinuity or to formulate a reasonable rule of thumb.

I have the feeling full-service/legacy airlines tend to have steeper price changes when specific fares become unavailable so, with extensive data, it might be possible to say more. My completely anecdotal experience suggests that the most dramatic increases might happen two weeks before departure but I have seen exceptions as well (and of course one-way and return prices do not necessarily move in sync).

Some search engines like ITA Matrix, Momondo, or Adioso can show the best price available for a given route over one or two months so you could have a look at that but it still does not fully address the question (each look-up only covers one route and since you can only query the prices available on a given day, time-until-departure and calendar date are confused in the data).