Is a p-value of 0.04993 enough to reject null hypothesis? In a Wilcoxon signed-ranks statistical significance test, we came across some data that produces a $p$-value of $0.04993$. With a threshold of $p < 0.05$, is this result enough to reject the null hypothesis, or is it safer to say the test was inconclusive, since if we round the p-value to 3 decimal places it becomes $0.050$?
 A: It lies in the eye of the beholder.
Formally, if there is a strict decision rule for your problem, follow it. This means $\alpha$ is given. However, I am not aware of any problem where this is the case (though setting $\alpha=0.05$ is what many practitioners do after Statistics101). 
So it really boils down to what AlefSin commented before. There cannot be a "correct answer" to your question. Report what you got, rounded or not.
There is a huge literature on the "significance of significance"; see for example the recent paper of one of the leading German statisticians Walter Krämer on "The cult of statistical significance - What economists should and should not do to make their data talk", Schmollers Jahrbuch 131, 455-468, 2011.
A: There are two issues here:
1) If you're doing a formal hypothesis test (and if you're going as far as quoting a p-value in my book you already are), what is the formal rejection rule?
When comparing test statistics to critical values, the critical value is in the rejection region. While this formality doesn't matter much when everything is continuous, it does matter when the distribution of the test statistic is discrete.
Correspondingly, when comparing p-values and significance levels, the rule is:

          Reject if $p\leq\alpha$

Please note that, even if you rounded your p-value up to 0.05, indeed even if the $p$ value was exactly 0.05, formally, you should still reject.
2) In terms of 'what is our p-value telling us', then assuming you can even interpret a p-value as 'evidence against the null' (let's say that opinion on that is somewhat divided), 0.0499 and 0.0501 are not really saying different things about the data (effect sizes would tend to be almost identical).
My suggestion would be to (1) formally reject the null, and perhaps point out that even if it were exactly 0.05 it should still be rejected; (2) note that there's nothing particularly special about $\alpha = 0.05$ and it's very close to that borderline -- even a slightly smaller significance threshold would not lead to rejection.
A: In light of the assumptions of your model, you should reject the null because dichotomizing claims based on hypothesis tests have clear epistemological and pragmatic functions.  But never forget that: “No isolated experiment, however significant in itself, can suffice for the experimental demonstration of any natural phenomenon; for the ‘one chance in a million’ will undoubtedly occur, with no less and no more than its appropriate frequency, however surprised we may be that it should occur to us. In order to assert that a natural phenomenon is experimentally demonstrable we need, not an isolated record, but a reliable method of procedure. In relation to the test of significance, we may say that a phenomenon is experimentally demonstrable when we know how to conduct an experiment which will rarely fail to give us a statistically significant result.” Fisher, R. A. (1935). The design of experiments. Oliver & Boyd.
In the pragmatic sense, you should reject. In the statistical sense, you need more data.
A: The 0.05 threshold is a hurdle that you have set for yourself in order to enforce a degree of self-skepticism about your alternative hypothesis. It somewhat weakens that self-skepticism if you change the definition of the threshold after seeing the result.  The real question is why you are performing an NHST, what do you think it tells you (probably not very much in most cases)?
The threshold should be set according to the nature of the experiment, so there is no one-true threshold anyway.  It would have been just as valid to set the threshold at 0.04992 (bit of an odd choice) before performing the NHST, so the difference isn't really meaningful (except in what it tells us about our self-skepticism).
You could always just report the p-value and let the reader draw their own conclusions (i.e. not reject or accept anything).
A: The answer is absolutely not. There is no "in the eye of the beholder", there is no argument, the answer is no, your data is not significant at the $p=0.05$ level.  (Ok, there is one way out, but its a very narrow path.)
The key problem is this phrase: "We came across some data...". 
This suggests that you looked at several other statistical hypothesis, and rejected them because they did not reach your significance level. You found one hypothesis that (barely) met your standard, and you are wondering whether it is significant. Unless your $p$ value accounts for such multiple hypothesis testing, it is overly optimistic.  Given that you are just three decimal points away from your threshold, considering even one additional hypothesis would surely push $p$ over the line.  
There is a name for this sort of statistical malfeasance: data dredging.  I'm ambivalent about reporting it in the paper as an interesting hypothesis; does it have some physical reason you expect it to hold?
There is, however, one way out. Perhaps you decided a priori to perform just this one test on just this one data set. You wrote that down in your lab notebook, in front of someone so that you could prove it later. Then you did your test. 
If you did this, then your result is valid at the $p=0.05$ level, and you can back it up to skeptics like me. Otherwise, sorry, it is not a statistically significant result.
