Normally a household fuse-box will have two fused lighting circuits (fused at 5A), one for the downstairs ring and one for the upstairs. This allows for a total lighting load on each ring of just over 1,000W. Given than a 100W bulb if sufficient for most rooms, 5A gives ample capacity in most applications.
When a bulb blows, often the filament within the bulb breaks and causes short circuit across the bulb element. Within the design of the bulb is a deliberate weak link (fuse), which should break when this happens (saving the 5A circuit fuse from operating).
You might just be unlucky that this weak link within the bulbs� has no operated as designed, but the 5A fuse box fuse is doing its job � protecting the circuit from an excessive current, due to a fault.
Many modern house installations use over-current trips instead of fuses. These tend to be more sensitive to this problem, resulting in the loss of the lighting circuit when a bulb blows � but they only need reset the trip, rather than replace a fuse.
As other have said, you should not increase the amperage of the fuse without having an electrician check out the wiring to be adequate.