Paraguay's economy is in flux, but it's still a relatively safe country. That said, don't wander around on your own late at night in Asunción and border towns. Several years ago there were reports of armed robbery on buses traveling at night. While this seems to have dissipated, snatching occurs frequently on local buses and muggings are still common. Don't display your valuables.
The Chaco is hostile and desolate with limited infrastructure - it is recommended that you go with guides. Poisonous snakes are common in certain areas, but mosquitoes are a likelier nuisance. Beware of strong currents when swimming in rivers.