Can we have chaotic motion due to the finite precision of our calculations?
The title question is a bit different from the one in the body of the post, so let's look at them separetelly:

Can we have chaotic motion due to the finite precision of our calculations?
Yes, Lorenz himself described the phenomenon, calling it computational chaos [Lorenz 1989]:
When one seeks approximate solutions of a set of differential equations by stepwise numerical integration, the choice of a time increment $\tau$ [...] may yield chaotic solutions, even when the true solutions approach limit cycles or fixed points.

cannot trust any of the predicted motions [?]
At least for hyperbolic systems actually yes, you can trust them. What gets you covered is the socalled shadowing theorem, which guarantees that, even though you're indeed not simulating the true trajectory of the initial condition you picked, there's a slightly different initial point whose trajectory remains arbitrarily close to the computergenerated trajectory. Check also this answer.
[Lorenz 1989] Computational chaosa prelude to computational instability, Physica D 35 (3), 1989, Pages 299317.
Yes, it is entirely possible that round off errors due to finite precision arithmetic can dramatically affect the outcome of computer simulations of nonlinear systems. In fact, one of the pioneers of modern chaos theory, Edward Lorenz, was inspired to study chaotic systems when he experienced this issue. Lorenz was running a weather simulation involving nonlinear differential equations on an early digital computer. When he tried to reproduce a scenario by entering initial values with three decimal places of precision, he found that the rerun diverged very quickly from the original output. Investigating the cause of this surprising behaviour, which Lorenz later described as the butterfly effect, led to the discovery of the Lorenz attractor.