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 $0.050$?
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.
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.
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.