If they are from the same provider:
If these connections act as a WAN, ideally you want to use some sort of routing protocol in my opinion (i.e.) BGP. You might also be able to use HSRP (or the similar options) if this is more like LAN of the provider offers it. Basically, talk to them about the options. HSRP for routers on the WAN side is not ideal, there are some problems that can still cause routing holes unless you put an extra needless layer of switches between your routers and your providers.
IP SLA or some sort of Small site Multihoming is a good middle option (Nice lab for this with Cisco here). Bigger option is to run BGP peering with both providers (not trivial). A less ideal option but perhaps good enough to use DNS trickery.
With whatever you chose, make sure you draw a picture (even better, build a lab) and start taking a single thing down at a time, a device, a connection, a service, etc... Then ask yourself, what happens in this scenario? If you don't put in this sort of effort, then I bet whatever you have done might be more likely to cause you to go down that the likelyhood of loosing a connection. Also, at the point you might be better of with a manual failover process.