Intermingling personal and work accounts is almost always a bad idea. I've seen colleagues get their personal accounts subpoenaed, have a firing result in loss of primary email/cell, or have really awkward communications show up in an IT audit. The overhead of maintaining separate accounts is very small (multi-profile browser, multi-account smartphone, etc.), and more than worth it given the liabilities. Beyond that,…
In particular, I won't set up my work email on my phone unlike many of my colleagues. For one, there have been stories of people's phones being wiped off thanks to an Exchange admin feature. And second, if the company I work for needs me to have access to my email at all times, they need to pay for my phone and my bill. (on top of it, it's typically forbidden by ToS to use your personal account for business purposes) For the same reason, I won't put my cell phone on my business cards or email signature (actually, I don't use email signatures…).
I also set up specific IM and Skype accounts for each company I work for. I don't want all my colleagues to know when I'm up or not, and I might not want to keep that contact after I'm gone. It really helps being harder to reach after hours.
In the case of Dropbox, I also create a new account with the added benefit that I can refer myself and get another 250MB for both accounts. (new email address, new computer) It's not very complicated to manage. I have a couple of shared folders for stuff I really need to access from both computers.