It is indeed a very debated concept, as this threshold is as arbitrary as it can get. Let's put it this way: if you consider 0.8 (usually 0.75) as your threshold you can say that 4 out of 5 of your resampled datasets produced the same result as the original data...does it make it a window open to the reality of what your are studying? I wouldn't say so, but at least you can say that something is going on there to push the results toward that direction.
I always use the bootstrap 1000 replications, because of reliability of results in this case.
It also depends on the dataset. If this is a phylogeny based on one gene it is quite straight forward.
However, if you have concatenated some sequences with different amounts of phylogenetically informative sites(PI sites) those "missing points" might be showing a different story. For instance in case of a hybridization. Phylogenies based on mtDNA or cpDNA can have a different topology than the nuclear DNA. And if they are concatenated together you might get low bootstrap values even though you got lots of PI sites.
But as mentioned by others, 50% is the typical cut-off for what to collapse(anything lower is worse than a coin flip), and 70+% is branches worth discussing.
I assume this is with Maximum Likelihood (or Maximum Parsimony). And would recommend a comparison with a phylogeny based on Bayesian Inference, which might give some support where ML or MP does not.
bootstrap can be set to 1000 replications if the dataset is not that big. For genome size data, bootstrap 500 or 100 is also good enough, I think. 70% bootstrap support can be think as reliable.
From the above answers it seems there are more problems than solutions.
Remember that the bootstrap is revealing the statistical consistency of your data with a hypothesis ( which is useful knowledge) but is not revealing the probability that your hypothesis is correct. People often confuse these two things.
It depends on what data you put in. If you build trees with primate data, the support for humans sharing a branch with chimpanzees vs gorillas is a good example. For most genes there is good support for ((human,chimp)gorilla) but for some genes there is good support for ((human, gorilla)chimp). So if you include dozens genes you might get anywhere from 70% to 100% bootstrap support for ((human,chimp)gorilla) depending on which genes exactly. If you leave out gorilla there will of course be 100 % support for ((human, chimp)orangutan) and if you leave out chimp there will be 100% support for ((human, gorilla)orangutan).
The point is, that the bootstrap support for a given node in a tree is one useful bit of information, but the usefulness of that bit of information depends on what data was put in, how densely sampled were related species, how distant are those species, how many genes were included, etc.
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