Why do ADSL router capacitors bulge after a few months?
I would not imply that ADSL router capacitors (always) bulge after a few months.  I see no plausible cause specific to ADSL routers generally, regardless of what happens in such and such a model.

Electrolytic capacitors are a relative weak point in any device.  With that said, they can last decades.


Appropriate manufacturer choice is important.  The capacitors shown in the photo in your linked repair article are from a second-rate manufacturer.
Thermal design is also important.  Even electrolytics from the best manufacturers (typically Japanese) will fail if they are allowed to get too hot. This can be due to bad original design, or user error (blocking ventillation, etc.)
ESR must also be considered. Some designs call for capacitors with especially low ESR.  If average caps are substituted, there will be trouble. One single supply chain cockup and a whole batch of products can be vulnerable. Also, most ADSL routers are consumer products with aggressive cost saving in their design.


Read about the historic capacitor plague here: https://en.wikipedia.org/wiki/Capacitor_plague.  This article provides plenty of information about failure modes for electrolytic capacitors.