Hi and welcome to the site - you're in good company as we've all gone through the exact same things you're describing. BPD is such a scripted illness it tends to play out almost exactly the same for everyone.. read the many other threads in these forums and you'll see they're identical.
It's hard to understand - and quite a shock - when that first breakup happens; going from happiness into sudden detachment and seemingly out of the blue for no reason. but BPD is an illness of high emotions and often the BPD sufferer doesn't know what they want from day to day.
is it possible for someone to have genuine feelings but struggle with the relationship and push the other person away?
Absolutely! Ironically, BPD causes the sufferer to react more strongly with the person they're closest too - you. This means you get their good and happy days but also their bad ones too. BPD people can also feel 'swamped and suffocated' by situations that wouldn't bother a non-BPD. They can be very happy to be in a relationship one day but then suddenly feel overwhelmed at all that's going on and suddenly withdraw - which can involve being seemingly blunt and uncaring to their partner.
BPD is an illness where everything is black and white and there is little middle ground, which leaves them unable to sit down and discuss things logically - as we could - so they tend to go to an extreme and just walk away. It doesn't mean they want to do this, it's simply their emotions running them.
Its not personal so try not to take it that way - though it is very hard not to when it happens. You're treating them great and they then just walk away like you've treated them badly. We've all been through this.
You can give your partner some space when these incidents happen as there's nothing to be gained from trying to push them into getting back - all here have tried that and it never works against a strongly-set BPD mind.
Still let them know you're there for them though, without being too pushy. Again, due to their black and white thinking we have to be careful of our responses; too much response and they'll feel overwhelmed and back off further, while too little response can make them think you're not interested in keeping them. You need to be a good balancer when with someone with BPD

Try not to be guilt-tripped too - you're doing nothing wrong, you're just with someone who, due to their illness, doesn't have any rules to play by and it can cause us to doubt ourselves and think we must be causing the problems.
Hope this helps a little, I'm sure many more will be along to comment and advise too. Feel free to keep posting - it does us all good to write it down and get it off our chest.