Are there any downsides of getting different insurance from different insurers? E.g. getting term from insurer A then CI from insurer B instead of getting CI as an add on to my term policy?

It's usually more straight forward to buy all insurances from 1 company but I agree that you may be able to save costs by getting policies from different companies since not all policies are made equal (in terms of $ or coverages).

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I personally am doing that and think there's no big disadvantage apart from the fact that you may have to familiarize yourself with how each companies do claiming in case your FA are not around when you need them most.

On the contrary, it is more likely to cost more if u buy 2 same coverage-type from 2 insurer. There will typically be 'discount' for larger coverage purchase. Assuming you are buying your coverage at once. But if u are bought from insurer A, buying from insurer B is not much difference.

There are circumstances/strategy that will require buying from different insurers. But normally, it is advisable to buy from same insurer. Easier for u to too. If u want to learn more do ping me.