Carla - the answer is .... it depends!
There are various different strategies for buying domains, and attempting to use them for SEO purposes.
The best case would be to negotiate privately with the owner of an existing website, and purchasing the domain and all of the contents of the domain from them. Businesses merge all the time, This is regular thing, and Google won't punish or look down on anyone for doing this. It will be clear that the domain changed hands, probably by updates to the name servers, and whois information, but since the website never expired or dropped from the registry, Google would probably assume this was a merger.
Also, if you do this, one of the best strategies for dealing with the domain would be to redirect individual pages to the most analogous page on your site, to take greatest advantage of the merged site.
It's also possible to buy up expired or expiring domains. There are different stages in the life cycle of domain name expiration where you can buy names. If you buy them in what's called 'pre-release' status, the domains will be transferred to your own account before the name dropped out of the registry, so it will maintain it's original creation date. You won't get the content with the domain, but this kind of domain purchase 'might' have some more life to it.
Finally, there are domains that have completely dropped out of the registry. The links from these won't have much more life to them.
But in general, if you can buy the domain and website content directly from the original owner, it should show up as a business merger (or even as 'rebranding' in your case, since your website is new), and should be a relatively safe tactic.