Natural enzymes can. There are several natural enzymes that break a part clots, such as nattokinase, lumbrokinase, serratiapeptidase, bromelain, papain and other proteolytic enzymes. These enzymes can be taken separate from food and help thin the blood and keep clots from forming as well. As such they will not work best in the acute setting. Anyone taking a blood thinning med needs to discuss this with their doctor.
We can but... Clot "busters" are very useful tools in certain situations; for example acute cerebrovascular accidents (or strokes) is restoring blood flow to the brain and minimizing to eliminating the damage. However, they must be used cautiously because they are not innocent substances and carry serious side effects.
Risk/benefit ratio. Clot-busting drugs are used to break clots apart (e.g. Pulmonary emboli) but this depends on a number of factors. These drugs are potent, high-dose blood thinners injected into the body so it travels all over the body, not just the site of the clot. The danger (and reason for strict guidelines) is it can cause bleeds elsewhere like the brain. This is weighed against the severity of the clot.