What you want just isn't possible on a small scale.
IP Addresses are purchased in blocks by companies with a hell of a lot of money and more importantly, infrastructure. When you get one, unless you are going direct to your RIR (Regional Internet Registry), you are only renting the IP.
Even if you then owned an IP by going direct to your RIR (which can be very expensive just to register... See ARIN, RIPE), I highly doubt any VPS provider will add a route to your IP as it will require a significant change to their own infrastructure.
The best thing you can do is to use a single provider that specialises in uptime/100% availability, it will be expensive, but, this is by far the cheapest way of going around it.
I'm updating this answer significantly as I have received so many emails asking for more info or help and approximate costs...
What you are asking for is hard and expensive. For just two VPS from a provider - I highly doubt you will anyone willing... But if someone does want to go down this route, it isn't impossible.
To understand routing across the internet, you need to know that there is more than IP routing alone. BGP is a routing protocol used by providers in order for them to say "My ips are here". To participate in BGP, you need an AS number, and to get an AS number and IPs, you need to register with a RIR (as explained above).
When participating in BGP, instead of buying internet as a service from a provider, you buy IP transit from a provider which allows you to publish your AS number/IPs
To use a metaphor (and I hope this does simplify it!)... BGP is you driving... An
"AS" number is the city you are driving to, IP blocks are the street, individual IPs are the houses.
So, you would have IP transit at multiple locations and you would then publish BGP routes from one, and if connectivity to one site fails, you publish the route to the other location and ips are routed there.
As for price, you will need to find a colocation package that allows you to have IP transit... If you shop around, minimum would probably be £300 a month for a quarter rack.
Next, the transit itself - around £5-£10 per Mb of IP transit when purchased at low totals.
I am not sure on all RIR fees as they all have different prices and structures (e.g. ARIN pay for how many IP blocks you have, starting at $500 for a /24 with a max price of $32,000, where as ripe is €1,800 for unlimited).
Lastly, you will need a server (any price!) and a router capable of BGP.
After you have done this, duplicate it, or even triplicate it! (Other than the RIR fees, that's a one off!).
And if you can't follow this, I'm always up for consultation ;)