I am not. At 1K the atoms are essentially “frozen” and if they are in a meta-stable state, they may remain there. With more kinetic energy in the systems, atoms can escape local potential minima and thus change conformation. As already suggested by @srtee, it is always a good idea to visualize your trajectory and have a close look.
As you are already suspecting, to make bonds rigid you do not need to also constrain angles. You only want to eliminate the fastest motion in the system, so you can use a larger timestep and that are the bond stretches involving hydrogen atoms (since hydrogen atoms are so much lighter than carbon). Angle bending in comparison is quite slow.
To better understand the error, you need to know that an angle constraint is internally converted to constraining three bonds that make a triangle formed by the constituent atoms of the angle. Now if that angle is close to 180 degrees, the triangle is very “flat” and thus it is no longer possible to come up with a suitable force constant for the two smaller bonds to “unflatten” the triangle, since those are nearly parallel to the third bond.