Yes, it actually is true.
The data on the hard drive is stored on very tiny magnetic cells which can take the 2 values of a bit: 0 or 1. By creating a sequence of such bits, the data (whether it's a photo, a game, an MP3, and so on) is created. Most people believe that the data has no physical weight, and they say that it's all virtual: it simply cannot be measured physically. Those people are wrong.
The inter-atomic bonds that make the silicon microchip produce energy fluctuations that may seem harmless. But the fluctuations can create a weight difference of over 10^-40 g/bit. Though, at this level, a full 1Tb HDD would be only a few grams heavier that an empty one.
Moreover, the atoms also absorb the energy from the data transfer, by heat and electricity, hence achieving more weight.
To sum up, data downloaded on the computer (actually, almost any kind of operation you perform) may increase its weight. Though, you may not even notice this, as the weight difference can't be more then 10 grams, even with the larger HDDs available.
My personal experience and knowledge