No, not in plane (Euclidean) geometry. But on the surface of a sphere, a quadrilateral bounded by 2 sets of great circles [one set has its intersecting points at what would be the 'poles', and the other set has its intersecting points on opposite sides of what would be the 'equator'] will have 4 acute interior angles.
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Yes as for example a rhombus is a 4 sided quadrilateral that has 2 equal opposite acute angles and 2 equal opposite obtuse angle and the 4 angles add up to 360 degrees.
There can be 4 different acute angles in an irregular 4 sided quadrilateral that all add up to 360 degrees.
It is a rhombus which also has 2 acute angles
a rectangle has 4 right angles, a parallelogram has 2 acute angles and 2 obtuse angles
There's no general rule. The only thing you can say about the angles ina quadrilateral is that all 4 of them always add up to 360 degrees.