Yes/No, depending on platter, and other restraints, and yes I am aware in general though various drives can do a lot in speeds, but these are often by using the cache available in the drive last I checked, when cache is saturated speeds go down.
Physical Harddrive discs have their limit simply by their nature, you have to remember sata 1 was 1.5 gbps and physical spinning discs couldn't even saturate that, and it isn't like hdd transfer rates have increased dramatically on physical spinning disk hdd's since then, the extra bandwidth afforded by sata 2 and above is really only an advantage to things that need to cross talk or high speed ssd's as well as the various communication overheads, also remember that writes are generally slower then reads.
Mechanical HDDs have poor random performance, but the sequential performance on anything that isn't positively ancient is fairly solid as it scales with areal density of the platters. The larger mechanical HDDs get, the faster they are at the same platter count because more sectors are passing under the heads at any given RPM. The 40 dollar WD Blue 5400rpm 2.5" drive that came with the last cheap laptop I bought has better sequential performance than the 10k rpm WD Raptors I had in 2004.
A five year old, 5400rpm laptop HDD can sustain reads and writes in excess of 100MB/s (800Mbps) and a modern 7200rpm desktop drive will push around 200MB/s (1600Mbps) on it's outer tracks. That's the physical performance of the media and will be achieved with no use of any cache at all. This is easily demonstrated with any HDD benchmark program, or by copying a large file from one drive to another. Performance on the innermost tracks can fall to about half of the outermost tracks, and fragmentation or other factors can reduce things further, but you aren't going to find a mechanical HDD that's the limiting factor in the download speeds of a 250Mbps connection unless you go way back. Even if one is getting the bulk of their advertised 1Gbps speeds, a modern, fast, mechanical drive that isn't full or being used by other disk heavy apps will usually not be the limiting factor.
Examples of the sequential performance of some modern mechanical HDDs (last two are 5400rpm, rest are 7200rpm):

The main advantage of consumer SSDs has never been sequential performance; an SATA SSD is only 2-4x as fast as a mechanical HDD in this regard. They can easily have hundreds of times the random access performance, which is a big part of responsiveness, and that's their main advantage.