Consider events $A, B$. Let $X,Y$ be random variables for which we want to determine a concentration inequality of the form $\mathbb{P}(|X-Y|\geq \varepsilon) \leq c \exp(-C)$ for some constants $c,C$ which may depend on $\varepsilon > 0$.
Now, assume we know that on $A$, we can bound $|X-Y| \leq |Z| + |W|$. Additionally, we know that on the event $B$ we have that $|Z| \leq \delta$ and $|W| \leq \rho$ for some constants $\delta, \rho > 0$.
Is there a way to combine these two known facts to get a general concentration result? Assume we know both $\mathbb{P}(A)$ and $\mathbb{P}(B)$. Proof strategies I have seen using bounds in probability never include two separate bounds, and I would like to know how to combine such results.