Entrepreneurs should be aware of two costs of taking money from angel investors (there are obvious benefits to consider, too).
Administrative Overhead (Time Sink): investor relations, for investors of all stripes, takes time by the CEO (but is very important). So, large numbers of angel investors can be a huge time sink for the CEO of a start-up (the highest number in any of my "Sherpa" companies was 37, and managing them was a nightmare). Most angels understand that, tied with money, time is the most valuable asset a start-up CEO has, so most angels act appropriately. But, quite often, inexperienced angels will take up a lot of the CEO's time with requests for calls, information, etc. For experienced angels, a $25K investment (the "atomic unit" of angel investing) is a small amount, and they realize the CEO doesn't have lots of time for every such angel; for some inexperienced angels (even wealthy ones), however, investing $25K (even into a $400K round) entitles them (they assume) to a lot of attention from the CEO. It's not universal, but it's also not unusual, for angels to be demanding in inverse relation to the size of their investment. Where the angel is an inexperienced start-up investor, it's definitely true.
Others have written about the above-referenced point, but it's worth re-emphasizing. I've seen it be a problem.
Unanimous Shareholder Vote: A lesser known, more subtle and nefarious, point is that even a small angel investor can cause problems in an M&A transaction because of the growing requirement that the shareholders of the selling company unanimously approve the transaction. There are a number of reasons for this, but the important thing here is to realize it's a trend.
This is a problem when either of two things happen:
(1) Unavailability: It's important to remember that most angels do their investing as a hobby, or other form of sidelight. This has several consequences, two of which are important to this topic: (a) they often are unavailable when their signature is needed on a document, and (a) they don't have "offices/assistants" who can run down a signature when needed.
When the angel is unavailable (my personal Sherpa deal nightmares were angels who were unavailable to sign because they were (a) trekking in Nepal for a month, and (b) sailing among the Galapagos Islands). Thus, even people of good intent can cause harm by not being available — important to remember that most "angels" aren't sitting around waiting for you to call.
This is bad enough, but it can get worse.
(2) Extortion: I have also encountered the situation where a small angel investor attempts to extort something extra out of the deal as a condition to tendering his signature. The unanimous shareholder vote requirement can give bad people leverage (one wouldn't think this, given it's a small town, but human greed is infinitely deep and many angels who not "in the community" don't consider their reputation).
Start-ups can't always raise money in an optimal way, of course, but, whenever you have choices, it's important to realize that angel investors, particularly in large numbers can impose considerable costs on your company in unexpected ways. You don't want a single angel (recipe for disaster), but you also don't want 37. Been there, done that. A word to the wise is sufficient.