Yes, you can!
If you wish to sign up for fibre broadband while continuing your existing DSL or cable broadband service, there are no technical issues. Fibre broadband uses the Netlink Trust/ OpenNet (NLT) Termination Point (TP) installed in your home and connects to the fibre optic Next Gen NBN infrastructure that is entirely separate from DSL or cable lines.
If you wish to run multiple fibre broadband services at once, it should be no problem as well. The NLT TP has 2 ports that allow for a minimum of 2 fibre broadband services running at the same time.
Do take note that activating the 2nd port of the TP will take around 4 to 8 weeks across all households in Singapore.