Why would a ship that is able to recycle all water consumed have trouble running out of water?

Unlike Galactica, many of the civilian ships were not equipped with effective water recycling system. Quote from your source:

Many of the ships like the Virgon Express were not made for long-term voyages - >and will have to tank off of us periodically.

This, possibly coupled with the sabotage disrupting (or stopping completely) the recycling process (see A J's answer) could lead to the depletion of the rest of the water supply in matter of weeks.

Why would a ship able to recycle all water consumed have trouble running out of water?

Well, it surely takes time to recycle the water. And you would need water for all necessities over that time. If you don't have enough water, how you are supposed to survive, given that water is the most important necessity for human and other living things.

Borrowed from the answer from other SE

With efficient recycling you can get by with a little more (there will always be some loss — and retention as well; quoting a different SF series, carbon-based life is "ugly bags of mostly water") than just enough water to keep people, animals, plants, etc. alive for several days; you need long enough for enough of the water consumed to get back into the ecosystem and be recycled. Lose enough water, though, and you could easily get into a situation where you don't have enough water for all necessities over the period needed for that water to get back through recycling — and water is more critical than food; you can if necessary go a week or so without food, but only a couple days without water. And many plants are more sensitive than humans.

(emphasis mine)

If we recycle all water in a self-sustainable way, and lose 60% of our water, the only way our loss is sustainable is if our initial water utilization was only 40% or less.

That is, we're in big trouble unless our ship was carrying 250% of its peak water demand.

This seems inefficient. If we can carry that much water, why worry about recycling so efficiently that we waste "not a drop"?

But if our water supplies are any less, then after losing 60%, we don't have enough to match peak demand: we'll need to ration, to lower that peak.

This will be uncomfortable, and there will be certain maintenance operations which cannot be rationed away: merely postponed. The food supply will suffer if irrigation is reduced for too long; sanitation will suffer if water for cleaning the ship is reduced too long; morale will deteriorate rapidly; etc.

How quickly this all becomes an issue, and how urgent that issue is, depends on lots of factors. But in the absence of life-threatening risk, it would certainly be the highest-priority issue on the ship.

Previous answers are spot-on, but there are other (perhaps even major) points of water loss that each overlooks that would cause loss even in a system with 100% recycling efficiency:

• People transferring from ship to ship. If an individual comes from another ship in the fleet and eats and/or drinks while aboard the Galactica, then departs, the quantity of water they've consumed is a net loss to the Galactica's reserves. While individually small, these losses would tend to accumulate over time, although they would be partially offset by people boarding the Galactica (perhaps even enough to roughly balance).

• Atmospheric loss. There is a large volume of air aboard Galactica, which contains water vapor. As air is lost, some of this water vapor is also lost and will be replaced through evaporation.

• Open-ended chemical processes. Human consumption is not the only drain on water; water is likely also used for other purposes, some of which may be open-ended (non-recycling).

• Food export. Galactica is shown during the series to be a major source of food, etc. for other ships in the fleet. Assuming that at least some of this is produced aboard Galactica (and not just prepackaged provisions), this would be a steady drain on her water resources.

• Absorption of atmospheric moisture. Many materials are hygroscopic, that is, they absorb moisture from the atmosphere, which is a net loss to the overall water supply. - Credit to @Tonny, who noted this in the comments.