No, short rests and long rests are mutually exclusive.
The PHB has the following to say about long rests:
If the rest is interrupted by a period of strenuous activity - at least 1 hour of walking, fighting, casting spells, or similar adventuring activity - the characters must begin the rest again to gain any benefit from it.
From this we can infer that if you are trying to take a long rest, you get no benefits from it until you complete the full rest. If sufficiently interrupted partway through - no matter how long you had been resting beforehand - you must begin the rest again to gain any benefit from it. There is no allowance here for getting the benefits of a short rest if you managed to go at least an hour before getting interrupted.
As written, you would have to first choose to have a short rest, wait out that hour, and then begin a separate long rest if you wanted to try and get the long rest's benefits. Luckily, if the rest interruption lasts less than an hour - which would certainly be the case for most combats - you can finish the current rest and get the full benefit at the end.
V2Blast's answer includes reference to comments from the designers which clarify that these are the rules as intended, or at least as subsequently interpreted by their writers.
I personally am not much a fan of this interpretation. A rest is a rest, and it doesn't make a lot of sense that how you feel after resting for an hour depends on whether you planned to only rest for an hour or were trying for a full night. Additionally, there is no requirement stated in the rules that the party must declare the kind of rest they are trying to have when they start to do so (the necessity to do so is only implied by the way that the designers think rests should work).
I would personally rule that: if you sit down to rest and make it at least an hour before being interrupted, you get the benefits of a short rest, and then if you continue resting to the full eight hours (and get your six hours of sleep), you get the benefits of a long rest.
Though you'd probably prepare a little differently if deliberately settling down for a long rest as opposed to a short one, the actual activity restrictions during short or long rests are almost identical save for that a long rest requires you to get six hours of sleep in the eight hours total - that leaves two hours in which you can do exactly what you would have been doing if you were taking a short rest, so they hardly seem exclusive to me. If you enforce separation between long and short rests, you're likely to have awkward logistics where wary players always have a short rest and then have a long rest, in order to get the short rest's benefits just in case they are interrupted during the night - and this mechanical complexity seems unnecessary and unrealistic. Just let a character rest!
It is important to note that this covers one continuous resting period (as the rules say, long and short rests are periods of "at least" the specified time) and so I would say you should only be able to take the benefits of a short rest once, after the first hour of the rest period; if you allow every hour of an eight-hour rest to count as an individual short rest as well as being a part of a long rest, you will probably run into some issues where players can abuse the mechanics.