We consider the Constrained-degree percolation model on the hypercubic lattice, $\mathbb L^d=(\mathbb Z^d,\mathbb E^d)$ for $d\geq 3$. It is a continuous time percolation model defined by a sequence, $(U_e)_{e\in\mathbb E^d}$, of i.i.d. uniform random variables in $[0,1]$ and a positive integer (constraint) $\kappa$. Each bond $e\in\mathbb E^d$ tries to open at time $U_e$; it succeeds if and only if both its end-vertices belong to at most $\kappa -1$ open bonds at that time. Our main results are quantitative upper bounds on the critical time, characterising a phase transition for all $d\geq 3$ and most nontrivial values of $\kappa$. As a byproduct, we obtain that for large constraints and dimensions the critical time is asymptotically $1/(2d)$. For most cases considered it was previously not even established that the phase transition is nontrivial. One of the ingredients of our proof is an improved upper bound for the critical curve, $s_{\mathrm{c}}(b)$, of the Bernoulli mixed site-bond percolation in two dimensions, which may be of independent interest.