It seems very likely that 3rd party support for the PS3 & 360 is going to gradually shift to the PS4 & Nextbox as we transition to next generation. The success of the PS3 & 360 for 3rd parties should be reason enough for them to embrace their successors. And considering the relatively low support for the WIi U, it doesn't look like Nintendo is going to attract a large portion of 3rd parties any time soon. That leaves the PS4 & Nextbox as the consoles that developers will flock to. They won't necessarily support next generation consoles exclusively. I think it'll be like last generation; developers will start with multi-generation titles and eventually transition towards next gen exclusives as they become more competent with the hardware.

If the PS4 and Nextbox do get heavy 3rd party support, are they guaranteed success? History has shown us that consoles that do receive heavy support from the majority of developers do see success. If this trend continues then the PS4 & Nextbox should be successful as well. However, some people argue that the market will see an unprecedented change which would offset this rule - I don't see the reasoning behind this, but some genuinely believe it's going to happen. Current generation gaming is rapidly declining and there's no other place console gamers can go to satisfy their needs. I see nothing that suggests that gamers won't transition in masses to the next generation consoles, as they've done every generation.

TL:DR

Are the PS4 and Nextbox going to receive heavy 3rd party support? If not, then why don't you think so? If so, then does that guarantee success?

PS1

Guarantee? No. High probability. Yes. The console that ends up selling the most units worldwide usually has the most content that appeals to both hardcore and casuals. Content is King. Always has been, and even with this gen it always will. Just look at all the blunders the PS3 went through during its initial days. The PS3 is still the most expensive console, and it's still 3rd worldwide. But its increase in recent sells over the competition can be attributed to its content. Never before a third place console has so many software offering as if it were 1st while outpacing the competition. Again, Content is King.

Good 3rd party support is a huge advantage, no doubt. Its not a recipe for sucess, or victory anyway, on its own though. The 360 and PS3 are the perfect examples of this, there are so many factors to decide hardware sales, as the Wii showed the world.
The biggest advantage of 3rd party support if the stability it offers, its probably the most stable sales incentive of them all in a longterm perspective, compared to pricing and economy, consumer awareness, percieved value and other factors such as hardware quality and online and media features.

It's almost guaranteed imo. EVERY system (EXCEPT Wii) that has won its gen has had the best 3rd party support. NES, SNES, PS1 and PS2 has had the lions share of 3rd party games, and they all won, so it stands to reason that having the best 3rd party support will secure the gen for you.

Yes. People don't care for 1st party exclusives much anymore, unless it's Nintendo.

3rd party exclusives, timed exclusive, and exclusive DLC of "AAA" major 3rd party titles will be the things that sway people to one console over another.

Yes.

Ill take that and I will raise you a PS3.

They literally  WILLED that system to success, and several companies went down in flames as result of those efforts.