Making the overall computation take a fixed amount of time does not plug indirect leaks. In particular, if the computation makes memory accesses at addresses that depend on secret values, then this will populate the memory caches for these addresses, and evict from the caches whatever data was using the same slots. The attacker may work out, after the computation, which previous data elements were evicted from cache, thereby leaking secret information. Crucially, the cryptographic computation itself took a fixed amount of time, but leakage occurs nonetheless.

This kind of leak still counts as a "timing attack" because the test for cache eviction is based on the time it takes to next access the relevant element (and this occurs _after_ the cryptographic computation, from other code). Notably, this can still be done remotely, possibly from a large distance, thanks to the efficiency of modern networking (this is what makes timing attacks special among side-channel leaks: power analysis, electro-magnetic emissions, sound... require the attacker to be in the physical vicinity of the target system, while timing attacks may be performed from hundreds of miles away). Demonstrations have been made with network access (over ethernet or optic fibre), and, in another kind of setup, when the attacker can run his own code on the same hardware, even with logical isolation (another process, or even in another VM that happens to run on some other cores of the same machine).

Thus, "true" constant-time code will make no memory access at an address that depends on secret data (but the data contents, of course, may be secret). Similarly, it won't make conditional jumps that depend on secret data (because of the cache accesses for loading the code, and also because of the jump prediction cache, both having been successfully exploited in lab demonstrations).