I wonder if I can claim from company Insurance and NTUC can pass on some of the savings to me?
If NTUC’s costs are lower then they’ll pass on some of their cost savings to you along with other policyholders. That’s an indirect benefit, to be sure, but why not?
Double claiming is illegal
Double (or duplicative) collection of payouts can be illegal. It’s a form of insurance fraud. But filing two (or more) claims with insurers is perfectly fine and common as long as you notify every insurer of the claims you are filing with the other insurers so that they can coordinate payouts. On the claim form (or electronic equivalent) you’ll probably see a question to that effect, but even if you don’t just make sure all the insurance companies you’re dealing with know about any overlapping claims, that’s all.
Generically speaking, some policies are explicitly secondary to others. In other cases the insurance companies figure it out, usually splitting the difference. As a policyholder you should end up with the most generous coverage among the policies.
You are not required to file multiple claims for the same loss (with cross-notification), as Moejoseph points out. Although I might disagree about which one insurer you should choose. Often the company-provided medical insurance has a lower annual limit but does not exclude pre-existing conditions. The latter aspect can be quite valuable, something you might want to “spend” a little more carefully. As another example, you might have a public hospital Integrated Shield plan and ordinarily get all you care from public medical providers, but you hold back the company-provided coverage for those rare events when you want to dabble in the private sector. “It depends,” but just remember that if you file two or more claims for the same loss (medical bills) you have to make sure all the insurers are aware of the others.