Answer:
Yes, if the triangles aren't necessarily both Euclidean. This may not be what you meant, and the other answers may answer your question more appropriately, but this seems a good opportunity to point out some more exotic scenarios that can exist.
Explanation:
One thing we could do is take triangles from different geometries. For instance, the sum of the angles of a triangle drawn on a sphere will always be greater than 180" role="presentation">180180 degrees. So you could draw one triangle on a flat plane, another on a sphere, make the side lengths match, and you will get different sized angles.