For any reasonably large application, and with limited development resources, using 3rd party libraries whenever possible is in my view pretty much the only way to go.
Usually it will simply not make any sense for you to try to do everything yourself; use existing libraries for everything peripheral, and focus your own development resources on your actual business problems.
Stick with libraries with common, widely-accepted licenses, and you should be fine in most cases: LGPL, ASL, BSD and derivatives, MIT, etc.
If your customers or managers are worried (or just interested), have your CTO or other senior tech person write an "Official Open Source Policy" for you dev team, which consists of "strict evaluation" of all libraries you use, as well as checking the licenses with a lawyer. As part of evaluating open-source libraries, investigate the vitality of each project, and keep a local copy of the sources for the version you use, just in case, even if you normally use the binary distribution.
Edit: Addendum about using commercial / closed-source libs:
The basic premise is the same: If the thing you need is peripheral enough, or time-consuming or difficult enough to build yourself, then yes, seriously consider buying it from a 3rd party vendor. (The decision of course involves a trade-off between various factors: time, your own dev resources, cash, how important it is, etc.)
But with closed-source libraries you should be extra careful about the licensing terms you're committing to. (Also, does the price you pay include support and upgrades; for how long? You don't want to find yourself stuck with an old, sucky version because upgrading would suddenly cost too much.) Specifically, if you're seriously committing to some library, make sure you will gain access to the source code of the library in case of the vendor going under. Such clauses are quite common when buying more expensive software, I believe. While googling around, I found an article talking about this; here's a quote:
[...] a licensee should attempt to include
in the license agreement a provision
providing the licensee with access to
the underlying source code held by the
licensor so long as the royalty or
license fees are paid. Alternatively,
the licensee can seek to include a
provision that permits direct access
to the source code by the licensor in
the event that the licensor fails to
or is unable to fulfill its
affirmative obligations to debug,
maintain, train, etc.
Finally, if the deal is big (financially, or if it's otherwise important to you), it will definitely help if you have people savvy in business and contracts, not just technology issues, doing the negotiations with the vendor.