Brief? It was allowed from 1994 to 2009, and the official reason for removing it in 2010 was team expenses (they had to move expensive speed-refueling machinery around the world, and highly trained pit staff to match)
However, it appears that it wasn't until 1983 that a team (Brabham) deliberately built a car with a tank too small to hold enough fuel to finish a race [2]. Brabham went on to win the 1983 championship and refuelling was banned from 1984.
[1]: http://en.wikipedia.org/wiki/1957_German_Grand_Prix [2]: http://www.formula1-dictionary.net/refueling.html
So using the historical argument, it's the refueling ban whose stint has been brief (1984 - 1993, and 2010 - ?)
Both mid-race refueling and the refueling rule have had brief stints.
http://www.youtube.com/watch?v=HdyuN_cUqjA
I don't think F1 had this problem but with some races the fuel is made with methanol and the flames can be impossible to see:
Here's another disturbing pitstop fire of Heikki Kovalainen leaving the pit stall with the fueling hose still attached and the fuel gets sprayed on his compatriot Kimi Räikkönen, who went on to finish the race in 5th spot. http://www.youtube.com/watch?v=OvgjhfQLTNk http://www.youtube.com/watch?v=Hzg_9UXo9s8
Trying to refuel 50 liters (15 gallons) or more in less than 10 seconds is bound to be dangerous.
Refueling is not allowed in Formula 1, it has only been allowed during two brief stints in the early 1980s and late 1990s. It was/is silly dangerous and there were a few quite nasty pitstop fire incidents.
One goal of the new regulations with the turbo, the regenerative braking and the fuel limit is to put emphasis on the efficiency of the engines. The engine manufacturer with the best efficiency is at an advantage.
There's a risk that this will introduce boring fuel saving periods to races. But on the other hand the fuel limit is the same for everyone. If you look at this from a different perspective, there will periods where you can crank up the boost and mixture to allow for more power at the cost of more than average fuel consumption.